#6595411

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 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)(19,27)(20,22)(21,25)(23,26)(24,31)(28,32)(29,30);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,30)(21,23)(22,32)(24,29)(26,27)(28,31);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,22,18,23,31,24,26)(19,28,32,27,21,30,29,25);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,15)(14,16)(17,21,23,19)(18,25,26,27)(20,30,31,28)(22,29,24,32);(1,4,5,3)(2,8,6,7)(11,15)(13,16)(17,22,23,24)(18,20,26,31)(19,29,21,32)(25,28,27,30)

Polycyclic presentation:

2^9|a^2=hi,c^2=fg,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,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, 6502607);