#6701411

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,24,28,31)(20,29,25,21)(26,30,32,27);(10,13)(11,16)(17,19,22,28)(18,24,23,31)(20,30,25,27)(21,26,29,32);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,20,22,25)(18,21,23,29)(19,26,28,32)(24,27,31,30);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(20,25)(21,29)(26,32)(27,30);(17,21,22,29)(18,25,23,20)(19,27,28,30)(24,32,31,26)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^e=hi

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, 6608607);