#5353759

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1659 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=gh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5260955);