#6840648

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1609 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=fg,e^2=ghi,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, 6747844);