#582067

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1918 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=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, 489263);