#6110894

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

1304 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=fghi,e^2=gh,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, 6018090);