#7361722

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(19,27)(20,26)(21,24)(23,31)(28,29)(30,32);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,28,30,23)(20,21,26,24)(27,29,32,31);(9,11,12,16)(10,13,14,15)(17,19,22,30)(18,23,25,28)(20,31,26,29)(21,32,24,27);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,26)(18,24,25,21)(19,29,30,31)(23,32,28,27);(2,5)(6,8)(10,14)(13,15)(18,25)(20,26)(23,28)(29,31)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7268918);