#6110982

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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

1292 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)(3,7,8,6)(9,10,13,16)(11,14,15,12)(19,25)(20,27)(22,28)(23,30)(34,35)(36,43)(37,45)(46,47);(9,11)(10,14)(12,16)(13,15)(19,25)(22,28)(26,31)(29,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(2,5)(6,7)(9,12,13,14)(10,15,16,11)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,3,4,8)(2,6,5,7)(9,11)(10,12)(13,15)(14,16)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fgi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6018178);