#4912341

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1377 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fh,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4819537);