#7146668

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

1323 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,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, 7053864);