#158804

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

2292 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)(7,11)(10,12)(15,19)(17,20)(21,22,24,25)(23,27,28,26)(29,30,32,35)(31,34,36,33);(6,9)(7,11)(15,19)(17,20)(22,25)(26,27)(29,31,32,36)(30,33,35,34);(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,26)(24,25)(27,28)(29,31)(30,34)(32,36)(33,35);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,17,18,20)(21,23)(22,26)(24,28)(25,27)(29,30)(31,34)(32,35)(33,36)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=fgi,d^2=gh,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, 66000);