#4861264

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

1378 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=ghi,e^2=fghi,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, 4768460);