#559414

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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

2014 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)(7,12)(8,11)(9,10)(13,14,16,17)(15,19,20,18)(21,22)(23,30)(24,25)(26,28)(27,29)(31,35)(32,33)(34,36);(6,9)(11,12)(14,17)(18,19)(21,23,27,24)(22,25,29,30)(26,32,34,31)(28,35,36,33);(1,2,4,3)(5,7,10,8)(6,11,9,12)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,25,29,30)(26,32,34,31)(28,33,36,35);(1,3,4,2)(5,8,10,7)(6,12,9,11)(13,15)(14,18)(16,20)(17,19)(21,25)(22,23)(24,29)(26,33)(27,30)(28,31)(32,36)(34,35);(5,9,10,6)(7,12,8,11)(13,14)(15,18)(16,17)(19,20)(21,26,27,34)(22,28,29,36)(23,31,24,32)(25,33,30,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

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, 466610);