#6842007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32);(9,11)(10,13)(12,16)(14,15)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32);(2,6)(7,8)(10,14)(11,16)(17,19,21,26,20,27,28,32)(18,22,25,29,24,23,31,30);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(19,27)(22,23)(26,32)(29,30);(1,4,5,3)(2,8,6,7)(17,19,28,32,20,27,21,26)(18,23,31,29,24,22,25,30)

Polycyclic presentation:

2^9|c^2=fh,d^2=fgi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6749203);