#6484592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1603 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)(12,16)(13,14)(17,18,24,30,23,28,22,27)(19,29,20,25,31,21,26,32);(2,5)(6,8)(10,14)(11,12)(17,19)(18,25)(20,22)(21,27)(23,31)(24,26)(28,32)(29,30);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,23,26)(18,21,28,29)(19,22,31,24)(25,27,32,30);(9,12,13,11)(10,16,14,15)(17,21)(18,26)(19,30)(20,28)(22,25)(23,29)(24,32)(27,31);(10,14)(15,16)(17,22,23,24)(18,27,28,30)(19,20,31,26)(21,32,29,25)

Polycyclic presentation:

2^9|a^2=gi,c^2=fg,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dhi,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6391788);