#585580

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2358 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)(6,7)(8,12)(14,17)(18,20)(22,27)(24,29)(28,33)(32,35);(6,7)(10,11)(15,19)(18,20)(21,22,26,27)(23,29,31,24)(25,33,34,28)(30,32,36,35);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,31)(22,24,27,29)(25,30,34,36)(28,32,33,35);(5,7,9,6)(8,11,12,10)(13,15,16,19)(14,18,17,20)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,25,26,34)(22,28,27,33)(23,30,31,36)(24,32,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=g,e^2=f,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, 492776);