#4463299

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
automorphism	Aut(G) order 4194304
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

1278 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)(5,8)(12,19)(16,21)(17,23)(22,24)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40)(41,42,46,47)(43,44,50,48)(45,53,54,49)(51,55,56,52);(1,2,4,6)(3,5,7,8)(10,14)(15,20)(16,21)(22,24)(26,30)(27,34)(32,36)(35,40)(42,47)(43,50)(49,53)(51,56);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40)(41,43,46,50)(42,48,47,44)(45,51,54,56)(49,55,53,52);(1,3)(2,5)(4,7)(6,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40)(41,44,46,48)(42,43,47,50)(45,52,54,55)(49,51,53,56);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)(41,45,46,54)(42,49,47,53)(43,51,50,56)(44,52,48,55)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=g,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4370495);