#5119548

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

1354 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)(11,12)(15,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34,37,38)(35,39,41,42)(36,44,46,40)(43,48,45,47);(1,3,6,5)(2,7,4,8)(10,14)(15,16)(20,28)(23,30)(26,32)(29,31)(34,38)(35,41)(40,44)(43,45);(1,4,6,2)(3,7,5,8)(9,10,13,14)(11,15,12,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35)(34,39)(36,45)(37,41)(38,42)(40,48)(43,46)(44,47);(9,11,13,12)(10,15,14,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,46)(45,47);(1,5,6,3)(2,7,4,8)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,36)(34,40)(35,43)(37,46)(38,44)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=fghi,b^2=hi,c^2=fh,d^2=fi,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, 5026744);