#6820831

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=ghi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6728027);