#6064260

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,5)(3,7)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(34,38)(39,41)(40,44)(46,47);(2,5)(6,8)(11,15)(13,16)(18,22)(20,28)(23,25)(26,32)(33,34,37,38)(35,41,43,39)(36,44,45,40)(42,46,48,47);(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,2,4,5)(3,6,7,8)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(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=fg,c^2=h,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5971456);