#5024499

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1520 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)(3,5)(4,7)(6,8)(9,10,12,13)(11,14,16,15)(17,18,21,20)(19,23,24,22)(26,30)(28,36)(31,33)(34,40);(1,3,6,7)(2,4,8,5)(10,13)(14,15)(26,30)(31,33)(32,37)(38,39);(1,4,6,5)(2,7,8,3)(9,10)(11,15)(12,13)(14,16)(17,19,21,24)(18,22,20,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,5,6,4)(2,7,8,3)(9,11)(10,14)(12,16)(13,15)(17,20,21,18)(19,22,24,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,c^2=gh,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4931695);