#158668

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

2216 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)(10,12)(14,17)(15,19)(21,22,25,23)(24,27,28,26)(30,31)(32,36);(2,4)(7,11)(10,12)(15,19)(18,20)(24,28)(26,27)(29,30,33,31)(32,34,36,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,23,25,22)(24,27,28,26)(29,31,33,30)(32,35,36,34);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,24)(22,26)(23,27)(25,28)(29,32)(30,34)(31,35)(33,36)

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=gh,d^2=ef,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, 65864);