#7172296

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,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, 7079492);