#5117965

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,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, 5025161);