#6029234

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

1240 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)(27,31)(28,30)(29,36)(32,37)(33,39)(34,35)(38,40)(42,43)(44,51)(45,53)(54,55);(2,5)(6,8)(12,19)(15,17)(20,24)(22,23)(26,28)(29,37)(31,35)(33,40)(41,42,46,43)(44,47,51,49)(45,50,53,48)(52,55,56,54);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27,30,34)(26,31,28,35)(29,33,37,40)(32,38,36,39)(41,43,46,42)(44,49,51,47)(45,50,53,48)(52,55,56,54);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,26)(27,31,34,35)(29,36,37,32)(33,38,40,39)(41,44)(42,47)(43,49)(45,52)(46,51)(48,54)(50,55)(53,56);(1,3)(2,6)(4,7)(5,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,32)(27,33)(28,36)(30,37)(31,38)(34,40)(35,39)(41,45,46,53)(42,48,43,50)(44,52,51,56)(47,54,49,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=fg,e^2=g,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, 5936430);