#392197

order	512
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14680
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3335 conjugacy classes of subgroups; only the 496 normal subgroups are shown.

normal subgroups only

This group has 3335 conjugacy classes of subgroups — too many to draw. Only its 496 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12,8,13,6,27)(3,14,18,30,19,38,17,9)(4,10,23,32,22,33,21,50)(5,11,26,34,25,35,24,51)(15,36,42,53,41,57,40,28)(16,37,45,54,44,58,43,29)(20,46,49,64,48,31,47,55)(39,59,62,52,61,56,60,63);(1,3,6,17,8,19,7,18)(2,9,12,14,13,30,27,38)(4,15,21,40,22,41,23,42)(5,16,24,43,25,44,26,45)(10,28,32,36,33,53,50,57)(11,29,34,37,35,54,51,58)(20,39,47,60,48,61,49,62)(31,52,55,56,46,63,64,59);(1,4)(2,10)(3,15)(5,20)(6,21)(7,23)(8,22)(9,28)(11,31)(12,32)(13,33)(14,36)(16,39)(17,40)(18,42)(19,41)(24,47)(25,48)(26,49)(27,50)(29,52)(30,53)(34,55)(35,46)(37,56)(38,57)(43,60)(44,61)(45,62)(51,64)(54,63)(58,59);(1,5)(2,11)(3,16)(4,20)(6,24)(7,26)(8,25)(9,29)(10,31)(12,34)(13,35)(14,37)(15,39)(17,43)(18,45)(19,44)(21,47)(22,48)(23,49)(27,51)(28,52)(30,54)(32,55)(33,46)(36,56)(38,58)(40,60)(41,61)(42,62)(50,64)(53,63)(57,59)

Polycyclic presentation:

2^9|a^2=h,b^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^c=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341661, #72341825, #72341830, …
maximal subgroup of	#72281262, #72281284, #72281337, …
maximal normal subgroup of	#72281262, #72281284, #72281337, …
maximal quotient of	#72343122

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299393);