#6437065

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^b=ei,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344261);