#6740991

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1201 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)(3,7)(4,8)(5,6)(9,10,14,17)(11,19,21,15)(12,16,13,18)(20,24,22,23)(25,26,29,33)(27,31,37,47)(28,39,41,32)(30,34,43,51)(35,53,54,45)(36,46,38,48)(40,56,55,49)(42,52,44,50);(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27,29,37)(26,31,33,47)(28,40,41,55)(30,36,43,38)(32,49,39,56)(34,46,51,48)(35,44,54,42)(45,52,53,50);(2,6)(7,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,35)(29,41)(30,44)(31,45)(33,39)(34,52)(36,40)(37,54)(38,55)(42,43)(46,49)(47,53)(48,56)(50,51);(1,3,5,4)(2,7,6,8)(10,17)(15,19)(16,18)(23,24)(28,41)(32,39)(35,54)(40,55)(42,44)(45,53)(49,56)(50,52);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,36)(30,43)(31,46)(34,51)(35,40)(37,38)(42,44)(45,49)(47,48)(50,52)(53,56)(54,55)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6648187);