#7459772

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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)(3,6)(4,5)(7,8)(25,26,30,36,31,37,29,35)(27,32,40,49,41,50,45,53)(28,42,43,52,44,34,46,33)(38,48,55,51,39,56,54,47);(2,5)(3,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,27,31,41)(26,32,37,50)(28,39,44,38)(29,45,30,40)(33,51,52,47)(34,48,42,56)(35,53,36,49)(43,54,46,55);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,28,31,44)(26,33,37,52)(27,38,41,39)(29,43,30,46)(32,47,50,51)(34,35,42,36)(40,54,45,55)(48,49,56,53);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,20)(19,24)(25,28)(26,34)(27,39)(29,46)(30,43)(31,44)(32,48)(33,36)(35,52)(37,42)(38,41)(40,54)(45,55)(47,53)(49,51)(50,56);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,29,31,30)(26,35,37,36)(27,40,41,45)(28,43,44,46)(32,49,50,53)(33,42,52,34)(38,54,39,55)(47,56,51,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7366968);