#5183014

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

1354 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)(11,14)(12,15)(13,16)(19,25)(20,27)(23,28)(24,29)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(9,11,12,16)(10,13,15,14)(18,22)(23,28)(24,29)(30,32)(34,38)(35,42)(36,45)(46,47);(9,11,12,16)(10,14,15,13)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,47)(46,48);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,15)(14,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fg,c^2=fg,d^2=h,e^2=fghi,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, 5090210);