#579693

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2158 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

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, 486889);