#950623

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

1776 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 857819);