#7273767

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1168 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=h,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7180963);