#4445857

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1360 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)(9,10,14,13)(11,16,15,12)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34,38,39)(35,36,41,44)(37,45,47,40)(42,46,48,43);(9,10,14,13)(11,12,15,16)(19,26)(23,25)(27,32)(30,31)(34,39)(35,41)(40,45)(42,48);(9,11)(10,12)(13,16)(14,15)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,46)(47,48);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,15,13,11)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30)(33,36)(34,35)(37,46)(38,44)(39,41)(40,48)(42,45)(43,47);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,16,15,12)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,47)(34,40,39,45)(35,42,41,48)(36,43,44,46)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fh,d^2=g,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4353053);