#6826816

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	#30396	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,16,12,14)(17,18)(19,20)(21,30)(22,28)(23,29)(24,27)(25,26)(31,32);(9,10)(11,14)(12,16)(13,15)(17,19,23,26)(18,25,29,20)(21,31,24,22)(27,32,30,28);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,24,32,23,25,21,28)(18,26,27,22,29,19,30,31);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,15,16)(17,21,23,24)(18,27,29,30)(19,31,26,22)(20,28,25,32);(9,12,13,11)(10,14,15,16)(17,22,21,26,23,31,24,19)(18,28,30,25,29,32,27,20)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fhi,d^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6734012);