#7171769

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

1223 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,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, 7078965);