#5730819

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,6,5)(3,8,4,7)(9,10,13,11)(12,15,16,14)(18,22)(19,25)(20,28)(30,32)(35,41)(36,43)(38,44)(39,46);(1,2)(3,8)(4,7)(5,6)(18,22)(20,28)(23,26)(27,31)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,3,6,4)(2,7,5,8)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4,6,3)(2,7,5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,37,41)(34,38,40,44)(36,42,43,47)(39,45,46,48);(1,5,6,2)(3,8,4,7)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36)(34,39)(35,42)(37,43)(38,45)(40,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=fghi,d^2=fgh,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, 5638015);