#6649002

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

Polycyclic presentation:

2^9|a^2=fi,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=dfhi,e^b=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, 6556198);