#587238

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1706 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:

(1,2,3,4)(6,10)(7,11)(13,14,18,17)(15,20,16,19)(21,22,25,26)(23,29,30,27)(24,32,33,28)(31,34,36,35);(5,6,9,10)(7,8,11,12)(13,15,18,16)(14,19,17,20)(22,26)(23,30)(24,33)(34,35);(6,10)(7,11)(13,14,18,17)(15,19,16,20)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(5,7)(6,8)(9,11)(10,12)(13,16,18,15)(14,19,17,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,35)(33,36);(5,8)(6,7)(9,12)(10,11)(13,17,18,14)(15,19,16,20)(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=fg,c^2=fh,d^2=fh,e^2=fgh,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, 494434);