#6131100

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

1256 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=ghi,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, 6038296);