#5794021

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1427 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=fghi,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5701217);