#158812

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

2308 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)(21,22,25,23)(24,27,28,26)(29,30)(31,33)(32,35)(34,36);(1,2,3,4)(6,9)(7,11)(15,19)(17,20)(24,28)(26,27)(30,31)(32,36);(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(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)(14,17)(16,19)(18,20)(21,24,25,28)(22,26,23,27)(29,32)(30,34)(31,35)(33,36)

Polycyclic presentation:

2^9|a^2=fh,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, 66008);