#678211

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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=f,d^2=fgi,e^2=g,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, 585407);