#6648794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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)(27,32)(28,41)(29,34)(30,44)(31,36)(33,42)(35,45)(37,51)(38,47)(39,48)(40,55)(43,50)(46,52)(49,54)(53,56);(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)(26,32)(28,37)(29,38)(30,40)(31,39)(33,46)(34,47)(35,49)(36,48)(41,51)(42,52)(43,53)(44,55)(45,54)(50,56);(11,17)(14,21)(18,19)(22,23)(25,28,31,29)(26,33,36,50)(27,37,39,38)(30,43,45,42)(32,46,48,56)(34,44,41,35)(40,53,54,52)(47,55,51,49);(2,5)(6,7)(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,31,28)(26,34,36,41)(27,38,39,37)(30,43,45,42)(32,47,48,51)(33,44,50,35)(40,53,54,52)(46,55,56,49);(1,3,4,8)(2,6,5,7)(10,16)(14,21)(15,24)(22,23)(27,39)(28,29)(32,48)(33,50)(34,41)(40,54)(42,43)(49,55)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=ei,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, 6555990);