#6594942

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1392 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18,22,24)(19,26,29,23)(20,30,25,21)(27,32,31,28);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,23,24,26)(20,31,25,27)(21,28,30,32);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,21,18,22,25,30,24)(19,27,28,26,29,31,32,23);(1,4,5,3)(2,8,6,7)(9,11)(10,13)(12,15)(14,16)(17,21,22,30)(18,20,24,25)(19,28,29,32)(23,31,26,27);(11,15)(13,16)(19,29)(23,26)(27,31)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6502138);