#6648881

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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

1226 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6556077);