#494273

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1853
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1035 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280077, #72280256, #72280998, …
maximal normal subgroup of	#72280077, #72280256, #72280998, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401469);