#158627

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

2224 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)(15,19)(16,17)(18,20)(21,22,25,24)(23,27,28,26)(29,30)(31,33)(32,34)(35,36);(2,4)(6,9)(7,11)(14,17)(15,20)(22,24)(23,28)(29,30,32,34)(31,35,36,33);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,28)(22,26,24,27)(29,30)(31,33)(32,34)(35,36);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,20)(17,19)(21,24,25,22)(23,27,28,26)(29,31,32,36)(30,33,34,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=efg,d^2=eh,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, 65823);