#719296

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1902 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)(5,6)(7,10)(8,9)(11,12)(15,16)(18,19)(21,22,25,23)(24,27,28,26);(5,7,11,9)(6,8,12,10)(13,14,17,20)(15,18,16,19)(21,23,25,22)(24,26,28,27);(5,8,11,10)(6,7,12,9)(13,15,17,16)(14,18,20,19)(21,24,25,28)(22,26,23,27);(1,2,3,4)(5,9,11,7)(6,8,12,10)(13,16,17,15)(14,19,20,18);(5,10,11,8)(6,7,12,9)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,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, 626492);