#7485149

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1269 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,35,29,34,43,32)(27,36,39,47,37,31,44,45)(28,33,42,50,41,51,54,48)(38,49,52,56,40,46,53,55);(1,2,4,5)(3,7,8,6)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,29,37)(26,31,34,36)(28,40,41,38)(30,39,43,44)(32,47,35,45)(33,49,51,46)(42,53,54,52)(48,55,50,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(27,37)(28,41)(30,43)(31,45)(33,50)(34,35)(36,47)(46,56)(48,51)(49,55)(52,53);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,41)(26,33,34,51)(27,38,37,40)(30,42,43,54)(31,46,36,49)(32,48,35,50)(39,52,44,53)(45,55,47,56);(1,3,4,8)(2,6,5,7)(10,18)(15,19)(16,17)(23,24)(26,34)(27,37)(28,41)(32,35)(39,44)(42,54)(46,49)(55,56)

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7392345);