#4407257

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1362 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)(11,15)(14,16)(17,18,21,22)(19,25,27,23)(20,24,29,30)(26,32,28,31)(33,34)(35,39)(36,38)(37,44)(40,46)(41,42)(43,48)(45,47);(9,10,13,12)(11,14,15,16)(18,22)(23,25)(24,30)(31,32)(34,36)(35,41)(40,44)(43,45);(9,11)(10,14)(12,16)(13,15)(17,19,21,27)(18,23,22,25)(20,28,29,26)(24,32,30,31)(33,35,38,41)(34,39,36,42)(37,45,46,43)(40,48,44,47);(1,2)(3,6)(4,5)(7,8)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,32)(29,30)(33,36,38,34)(35,42,41,39)(37,44,46,40)(43,48,45,47);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,16,15,14)(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=f,d^2=h,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4314453);