#6700929

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 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)(17,18,23,22)(19,24,29,32)(20,25,21,26)(27,31,28,30);(10,13)(11,16)(17,19,23,29)(18,24,22,32)(20,30,21,31)(25,27,26,28);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,20,23,21)(18,25,22,26)(19,27,29,28)(24,31,32,30);(1,3,4,7)(2,6,5,8)(10,13)(14,15);(17,21,23,20)(18,26,22,25)(19,28,29,27)(24,30,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6608125);