#159678

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

2140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(6,9)(7,11)(13,14,17,16)(15,18,19,20)(21,22)(23,27)(24,25)(26,28)(30,33)(34,35);(2,4)(7,11)(10,12)(14,16)(18,20)(22,24)(23,28)(29,30,32,33)(31,35,36,34);(5,6,8,9)(7,10,11,12)(13,15,17,19)(14,18,16,20)(21,23)(22,26)(24,27)(25,28)(29,30)(31,35)(32,33)(34,36);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,18,19,20)(21,24,25,22)(23,26,28,27)(29,31,32,36)(30,34,33,35)

Polycyclic presentation:

2^9|a^2=eg,b^2=fh,c^2=ef,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, 66874);