#7089862

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#24275	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,21)(22,24)(23,26)(27,28)(29,32)(30,31);(9,10)(11,15)(12,14)(13,16)(18,24)(20,26)(21,22)(25,27)(28,31)(30,32);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,23,29)(18,25,26,32)(20,30,24,27)(21,31,22,28);(11,16)(13,15)(17,20,23,24)(18,21,26,22)(19,27,29,30)(25,28,32,31);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,20,26,24)(19,28,29,31)(25,27,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997058);