#3335201

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2279 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3242397);