#5783927

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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

1386 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)(17,18,21,22)(19,23,26,29)(20,24,27,30)(25,31,28,32)(33,34,38,36)(35,41,42,39)(37,44,45,40)(43,46,48,47);(1,3,4,7)(2,5,6,8)(10,13)(14,16)(18,22)(20,27)(23,29)(25,28)(35,42)(37,45)(39,41)(40,44);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,30)(28,32)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(9,11)(10,14)(12,15)(13,16)(17,19,21,26)(18,23,22,29)(20,28,27,25)(24,32,30,31)(33,36,38,34)(35,39,42,41)(37,44,45,40)(43,46,48,47);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,27)(18,24,22,30)(19,25,26,28)(23,31,29,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5691123);