#7135096

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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
prime radical	#45319	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,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, 7042292);