#6652352

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

1193 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)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,33)(29,32)(30,34)(35,42)(36,45)(37,44)(38,46)(39,48)(40,50)(41,47)(43,51)(49,54)(52,56)(53,55);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,31)(28,39)(29,38)(30,36)(32,46)(33,48)(34,45)(35,43)(37,40)(41,53)(42,51)(44,50)(47,55)(49,56)(52,54);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,32)(27,36)(29,42)(31,43)(33,47)(34,35)(37,39)(41,54)(44,55)(45,46)(48,56)(49,50);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,30,40)(26,33,34,50)(27,37,36,39)(29,41,42,54)(31,44,45,48)(32,47,35,49)(38,52,51,53)(43,55,46,56);(1,3,4,8)(2,6,5,7)(10,18)(15,24)(16,17)(22,23)(26,34)(27,36)(28,40)(32,35)(38,51)(41,54)(44,48)(55,56)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=hi,d^2=h,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,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, 6559548);