#759030

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=gh,d^2=fi,e^2=fhi,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, 666226);