#589934

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fgi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 497130);