#4512435

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1314 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)(6,7)(11,15)(13,16)(18,22)(19,25)(20,27)(30,32)(33,34,38,36)(35,41,42,39)(37,44,45,40)(43,46,48,47);(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,29)(26,31)(34,36)(35,42)(37,45)(46,47);(1,2,4,5)(3,6,8,7)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,42)(34,39,36,41)(37,43,45,48)(40,46,44,47);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,34)(35,39,42,41)(37,44,45,40)(43,46,48,47);(1,3,4,8)(2,6,5,7)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgi,c^2=g,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 4419631);