#641663

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2102 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)(6,8)(7,11)(14,18)(16,20)(21,22,25,24)(23,27,28,26);(5,6,9,8)(7,10,11,12)(13,14,17,18)(15,16,19,20)(23,28)(26,27);(7,11)(10,12)(14,18)(15,19)(21,23,25,28)(22,26,24,27);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,15,17,19)(14,16,18,20);(5,8,9,6)(7,12,11,10)(13,16)(14,19)(15,18)(17,20)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 548859);