#946762

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

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

2322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fh,d^2=fi,e^2=f,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, 853958);