#7090765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1636 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,21,24,26,22,32)(19,27,29,20,30,31,28,25);(9,10)(11,15)(12,14)(13,16)(17,19,24,30)(18,25,26,20)(21,31,32,27)(22,29,23,28);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,24,25)(18,19,26,30)(21,29,32,28)(22,27,23,31);(11,16)(13,15)(17,21,22,18,24,32,23,26)(19,20,28,27,30,25,29,31);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,21,26,32)(19,28,30,29)(20,31,25,27)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997961);