#6719803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1240 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6626999);