#868699

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

2062 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=ghi,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 775895);