#6911626

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907945
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=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, 6818822);