#159649

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

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, 66845);