#159830

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

2348 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)(10,11)(13,14,17,16)(15,19,20,18)(21,22,25,27)(23,24,26,28);(2,4)(5,6,8,7)(14,16)(18,19)(23,26)(24,28);(6,7)(9,10,12,11)(13,15,17,20)(14,18,16,19)(21,23,25,26)(22,24,27,28);(1,2)(3,4)(5,7,8,6)(9,11,12,10)(13,16,17,14)(15,18,20,19)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=fg,d^2=f,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, 67026);