#6846915

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1786 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,12,15)(17,18)(19,26)(20,28)(21,25)(22,23)(24,27)(29,32)(30,31);(1,3)(2,5)(4,8)(6,7)(11,12)(14,15)(17,19,21,22)(18,23,25,26)(20,27,29,30)(24,28,31,32);(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,28)(25,31)(26,32);(2,6)(5,7)(18,25)(23,26)(24,31)(28,32);(3,8)(5,7)(9,12,13,11)(10,15,16,14)(20,29)(24,31)(27,30)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi

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, 6754111);