#6034524

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fgi,d^2=g,e^2=h,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, 5941720);