#5805053

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fghi,c^2=fghi,d^2=g,e^2=fgh,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, 5712249);