#6844076

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1325 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)(9,10,12,14)(11,15,16,13)(17,18,23,22)(19,24,28,31)(20,25,30,21)(26,29,32,27);(9,11)(10,13)(12,16)(14,15)(17,19,23,28)(18,24,22,31)(20,29,30,27)(21,26,25,32);(2,6)(7,8)(11,16)(13,15)(17,20,18,25,23,30,22,21)(19,26,31,27,28,32,24,29);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(20,30)(21,25)(26,32)(27,29);(1,4,5,3)(2,8,6,7)(17,21,22,30,23,25,18,20)(19,27,24,26,28,29,31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fhi,d^2=g,e^2=fgh,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^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, 6751272);