#528746

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=gh,e^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

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, 435942);