#6550010

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 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)(4,8)(5,6)(17,18,22,25,23,24,21,31)(19,20,28,29,26,30,27,32);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,27)(22,28)(23,26)(24,30)(25,29)(31,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,19)(21,29)(22,32)(23,30)(24,26)(25,27)(28,31);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(18,24)(19,26)(25,31)(27,28);(11,15)(14,16)(17,21,23,22)(18,25,24,31)(19,27,26,28)(20,29,30,32)

Polycyclic presentation:

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

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, 6457206);