#5078894

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1390 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fghi,c^2=fgh,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 4986090);