#4523523

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

1296 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)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(1,2,4,5)(3,7,8,6)(18,22)(19,26)(20,29)(30,31)(34,38)(36,45)(39,41)(42,48);(1,2,4,5)(3,6,8,7)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,3,4,8)(2,6,5,7)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=hi,d^2=gi,e^2=g,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, 4430719);