#511966

order	512
nilpotency class	4
isoclinism	isoclinic class of #511966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32930
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

948 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 184 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341926
maximal subgroup of	#72320061, #72341926, #72342000, …
maximal normal subgroup of	#72320061, #72341926, #72343953, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419162);