#7019803

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,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, 6926999);