#553202

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×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

3302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(9,10,13,14)(11,15,17,18)(12,20,21,16)(19,24,23,22)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(2,4)(5,6,7,8)(10,14)(11,17)(16,20)(19,23)(25,26,29,32)(27,30,35,33)(28,31,37,36)(34,38,40,39);(1,2,3,4)(5,6)(7,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 460398);