#4783776

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1632 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,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, 4690972);