#7143432

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(19,24)(20,30)(21,33)(22,36)(23,27)(25,28)(26,35)(29,42)(31,45)(32,38)(34,39)(37,41)(40,46)(43,48)(44,47);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,23,22)(18,24,27,36)(20,28,32,37)(21,29,35,34)(25,38,41,30)(26,39,33,42)(31,43,47,46)(40,45,48,44);(9,10)(11,14)(12,13)(15,16)(17,20,23,32)(18,25,27,41)(19,28,22,37)(21,31,35,47)(24,38,36,30)(26,40,33,48)(29,43,34,46)(39,45,42,44);(3,8)(5,7)(10,13)(14,16)(17,21)(18,26)(19,29)(20,31)(22,34)(23,35)(24,39)(25,40)(27,33)(28,43)(30,44)(32,47)(36,42)(37,46)(38,45)(41,48);(9,11,12,15)(10,14,13,16)(17,19,23,22)(18,24,27,36)(20,28,32,37)(21,34,35,29)(25,38,41,30)(26,42,33,39)(31,46,47,43)(40,44,48,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,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, 7050628);