#6556884

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
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) = 256

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

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

Polycyclic presentation:

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

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, 6464080);