#5602202

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1316 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=h,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

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, 5509398);