#6966150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#37845	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1605 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)(17,18,24,32,22,26,23,27)(19,28,21,31,29,25,30,20);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,22,29)(18,25,26,28)(20,27,31,32)(21,23,30,24);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,31)(18,21,26,30)(19,27,29,32)(23,28,24,25);(2,6)(5,7)(10,13)(14,16)(17,21)(18,20)(19,23)(22,30)(24,29)(25,27)(26,31)(28,32);(9,11,12,15)(10,14,13,16)(18,26)(19,29)(21,30)(27,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6873346);