#769883

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2926 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)(13,14,18,19)(15,20,21,16)(17,25,27,22)(23,24,26,28);(9,10,11,12)(14,19)(16,20)(22,25)(24,28);(5,6)(7,8)(9,10)(11,12)(13,15,18,21)(14,20,19,16)(17,26,27,23)(22,24,25,28);(1,2,3,4)(13,16,18,20)(14,21,19,15)(17,24,27,28)(22,26,25,23);(13,17,18,27)(14,22,19,25)(15,23,21,26)(16,24,20,28)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

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, 677079);