#6034530

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

1236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

(3,7)(6,8)(9,10,14,11)(12,17,19,15)(13,18,21,16)(20,22,24,23)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38)(41,42)(43,47)(44,46)(45,49)(48,51)(50,54)(52,53)(55,56);(2,5)(6,8)(12,19)(15,17)(20,24)(22,23)(26,30)(28,36)(31,33)(34,40)(41,43,46,51)(42,47,44,48)(45,50,53,56)(49,54,52,55);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)(41,43)(42,48)(44,47)(45,50)(46,51)(49,55)(52,54)(53,56);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39)(41,44,46,42)(43,48,51,47)(45,52,53,49)(50,55,56,54);(1,3,4,7)(2,6,5,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)(41,45,46,53)(42,49,44,52)(43,50,51,56)(47,54,48,55)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=gi,d^2=g,e^2=fh,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, 5941726);