#586704

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2102 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,e^2=fh,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, 493900);