#4465605

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
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

1352 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)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,34)(35,39)(36,38)(37,44)(40,45)(41,43)(42,48)(46,47);(9,10,12,14)(11,13,16,15)(18,22)(23,25)(24,29)(30,31)(34,36)(35,41)(40,44)(42,47);(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,38,41)(34,39,36,43)(37,42,45,47)(40,46,44,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36,38,34)(35,39,41,43)(37,44,45,40)(42,46,47,48);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,13,14,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,45)(34,40,36,44)(35,42,41,47)(39,46,43,48)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=f,d^2=gh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,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, 4372801);