#155511

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2282 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fgi,d^2=efg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 62707);