#582010

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

1990 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=f,e^2=h,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, 489206);