#582617

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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

1954 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)(6,8)(7,11)(14,19)(15,17)(21,22,26,30)(23,24,28,27)(25,33,34,29)(31,35,36,32);(5,6,9,8)(7,12,11,10)(13,14,18,19)(15,16,17,20)(21,23)(22,27)(24,30)(25,31)(26,28)(29,35)(32,33)(34,36);(1,3,4,2)(5,6,9,8)(7,12,11,10)(13,15,18,17)(14,20,19,16)(21,23)(22,24)(25,31)(26,28)(27,30)(29,32)(33,35)(34,36);(5,7,9,11)(6,10,8,12)(13,16,18,20)(14,15,19,17)(21,24)(22,28)(23,30)(25,32)(26,27)(29,36)(31,33)(34,35);(5,8,9,6)(7,12,11,10)(13,17,18,15)(14,20,19,16)(21,25,26,34)(22,29,30,33)(23,31,28,36)(24,32,27,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 489813);