#7623240

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,27)(13,19)(14,35)(15,37)(16,21)(18,31)(20,29)(22,46)(23,39)(24,40)(25,53)(26,54)(28,55)(30,56)(32,63)(33,58)(34,45)(36,64)(38,49)(41,47)(42,59)(43,61)(44,62)(48,69)(50,72)(51,67)(52,66)(57,60)(65,71)(68,70);(2,5)(6,8)(9,11)(10,17)(12,28)(13,24)(14,25)(15,26)(16,23)(18,42)(19,40)(20,41)(21,39)(22,33)(27,55)(29,47)(30,50)(31,59)(32,48)(34,52)(35,53)(36,51)(37,54)(38,44)(43,68)(45,66)(46,58)(49,62)(56,72)(57,65)(60,71)(61,70)(63,69)(64,67);(9,12,29,60,15,32,36,62,16,33,34,61,13,30,14,31)(10,18,35,56,19,43,45,58,21,44,64,63,37,57,20,27)(11,22,47,70,26,50,51,59,23,28,52,71,24,48,25,49)(17,38,53,69,40,65,66,55,39,42,67,72,54,68,41,46);(1,3,4,7)(2,6,5,8)(11,23)(17,39)(22,28)(24,26)(25,51)(38,42)(40,54)(41,66)(46,55)(47,52)(48,50)(49,59)(53,67)(65,68)(69,72)(70,71);(9,13,16,15)(10,19,21,37)(11,24,23,26)(12,30,33,32)(14,34,36,29)(17,40,39,54)(18,43,44,57)(20,35,45,64)(22,48,28,50)(25,52,51,47)(27,56,58,63)(31,61,62,60)(38,65,42,68)(41,53,66,67)(46,69,55,72)(49,71,59,70)

Polycyclic presentation:

2^9|c^2=gh,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,c^b=ci,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530436);