#6487351

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1231 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,32,38)(27,39,43,50)(28,44,45,53)(29,35,47,54)(30,37,31,36)(33,42,51,41)(34,46,52,40)(48,56,49,55);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,42,32,43,30,41)(26,33,37,50,38,51,36,39)(28,29,40,49,45,47,46,48)(34,35,44,56,52,54,53,55);(11,17)(14,21)(18,19)(22,23)(25,28,31,40,32,45,30,46)(26,34,37,44,38,52,36,53)(27,29,42,49,43,47,41,48)(33,35,50,56,51,54,39,55);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29)(26,35)(27,40)(28,41)(30,48)(31,49)(32,47)(33,44)(34,39)(36,55)(37,56)(38,54)(42,45)(43,46)(50,52)(51,53);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(28,45)(29,47)(34,52)(35,54)(40,46)(44,53)(48,49)(55,56)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394547);