#5595088

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=g,c^2=h,d^2=f,e^2=ghi,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, 5502284);