#392612

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14693
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3325 conjugacy classes of subgroups; only the 416 normal subgroups are shown.

normal subgroups only

This group has 3325 conjugacy classes of subgroups — too many to draw. Only its 416 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72342071
maximal subgroup of	#72342071, #72342076, #72342963, …
maximal normal subgroup of	#72342071, #72342076, #72342963, …
maximal quotient of	#72287524

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299808);