#6757249

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,32,22,26,21,27)(19,20,28,30,25,24,29,31);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,22,25)(18,24,26,20)(21,29,23,28)(27,30,32,31);(9,11,12,15)(10,14,13,16)(17,20,22,24)(18,25,26,19)(21,30,23,31)(27,29,32,28);(2,6)(5,7)(10,13)(14,16)(18,26)(19,25)(27,32)(28,29);(3,8)(5,7)(17,21,22,23)(18,27,26,32)(19,28,25,29)(20,30,24,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6664445);