#158809

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
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

2460 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=e,c^2=fi,d^2=fgh,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 66005);