#4912268

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1294 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fh,d^2=gh,e^2=ghi,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, 4819464);