#7605776

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518904
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=h,d^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh

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, 7512972);