#7144086

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7051282);