#684081

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1882 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,32)(29,30);(1,3,5,4)(2,6,7,8)(9,10,13,12)(11,14,15,16)(18,20)(22,23)(26,30)(27,32);(9,11)(10,14)(12,16)(13,15)(17,18,21,20)(19,22,24,23)(25,27)(26,28)(29,32)(30,31);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,24)(18,22,20,23)(25,28)(26,27)(29,31)(30,32);(1,2)(3,8)(4,6)(5,7)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,23,24,22)(25,26)(27,28)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=gi,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 591277);