#6565976

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(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);(1,3,5,4)(2,6,7,8)(9,11,13,18)(10,14,16,17)(12,19,20,24)(15,21,22,23)(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);(11,18)(14,17)(19,24)(21,23)(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);(1,4,5,3)(2,6,7,8)(12,20)(15,22)(19,24)(21,23)(27,37)(31,35)(36,52)(38,54)(45,49)(46,50)(51,56)(53,55);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6473172);