#7334856

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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

1386 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7242052);