#4973904

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

1548 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fgi,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4881100);