#7623281

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622982
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#60717	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530477);