#6595100

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 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)(17,18,23,26)(19,24,22,20)(21,25,31,32)(27,30,29,28);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,26,20)(21,29,31,27)(25,28,32,30);(11,15)(13,16)(17,20,22,18,23,24,19,26)(21,30,29,32,31,28,27,25);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,15)(14,16)(17,21,23,31)(18,25,26,32)(19,27,22,29)(20,28,24,30);(1,4,5,3)(2,8,6,7)(11,15)(13,16)(21,31)(25,32)(27,29)(28,30)

Polycyclic presentation:

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

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, 6502296);