#6594841

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,28,30,20)(21,25,29,32)(22,26,24,31);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,30)(18,20,27,28)(21,24,29,22)(25,26,32,31);(11,15)(13,16)(17,20,24,32,23,28,22,25)(18,21,26,19,27,29,31,30);(1,4,5,3)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,21,23,29)(18,25,27,32)(19,22,30,24)(20,31,28,26);(11,15)(13,16)(17,22,23,24)(18,26,27,31)(19,29,30,21)(20,25,28,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,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, 6502037);