#7623221

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622979
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1268 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:

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

Polycyclic presentation:

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

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, 7530417);