#6694594

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

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

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, 6601790);