#7205433

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,32)(28,34)(29,35)(30,33)(31,36)(37,46)(38,47)(39,45)(40,48)(41,50)(42,49)(43,52)(44,51)(53,55)(54,56);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,31,40)(26,32,36,48)(28,41,43,37)(29,38,30,39)(33,45,35,47)(34,50,52,46)(42,54,44,53)(49,56,51,55);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(29,30)(32,45)(34,49)(35,36)(38,39)(42,44)(46,55)(47,48)(50,56)(51,52)(53,54);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,31,43)(26,34,36,52)(27,37,40,41)(29,44,30,42)(32,46,48,50)(33,49,35,51)(38,54,39,53)(45,55,47,56);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(25,29,31,30)(26,35,36,33)(27,38,40,39)(28,42,43,44)(32,47,48,45)(34,49,52,51)(37,53,41,54)(46,55,50,56)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,e^c=ei,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, 7112629);