#4757784

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1447 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)(6,8)(10,14)(11,16)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(9,10,13,14)(11,15,16,12)(18,22)(23,25)(24,28)(31,32)(36,44)(40,45)(42,48)(46,47);(10,14)(12,15)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,36,38,44)(34,40,37,45)(35,42,43,48)(39,46,41,47);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=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, 4664980);