#5069233

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1414 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=f,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 4976429);