#5578321

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

1308 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)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,25,27,23)(20,24,29,28)(26,31,32,30)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(11,15)(13,16)(18,22)(23,25)(24,28)(30,31)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(9,11,12,15)(10,13,14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,2)(3,5)(4,6)(7,8)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=gh,d^2=f,e^2=i,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, 5485517);