#587637

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

2058 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)(7,11)(8,10)(9,12)(14,16)(18,20)(21,22)(23,24)(25,33)(26,27)(28,31)(29,34)(30,35)(32,36);(1,3,4,2)(5,7,10,9)(6,11,8,12)(15,19)(18,20)(22,27)(24,28)(25,34)(30,36);(5,6,10,8)(7,12,9,11)(13,14,17,16)(15,18,19,20)(21,23,26,31)(22,28,27,24)(25,30,34,36)(29,35,33,32);(5,8,10,6)(7,11,9,12)(13,15,17,19)(14,18,16,20)(21,24,26,28)(22,23,27,31)(25,32,34,35)(29,30,33,36);(5,9,10,7)(6,11,8,12)(13,16,17,14)(15,20,19,18)(21,25)(22,29)(23,30)(24,32)(26,34)(27,33)(28,35)(31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=g,e^2=fg,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, 494833);