#6844058

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×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 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)(17,18,22,23)(19,24,29,32)(20,28,31,25)(21,26,30,27);(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,30)(21,31)(22,29)(23,32)(25,26)(27,28);(2,6)(7,8)(11,16)(13,15)(17,20,18,25,22,31,23,28)(19,27,32,21,29,26,24,30);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,18,22,23)(19,24,29,32)(20,25,31,28)(21,27,30,26);(1,4,5,3)(2,8,6,7)(17,21,18,26,22,30,23,27)(19,28,32,20,29,25,24,31)

Polycyclic presentation:

2^9|a^2=fi,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,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, 6751254);