#7623237

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

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,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, 7530433);