#7170288

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

1420 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)(10,13)(11,15)(17,18,19,26,22,27,30,23)(20,31,34,41,33,40,28,24)(21,35,29,38,37,25,36,42)(32,45,46,39,47,44,43,48);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,22,30)(18,23,27,26)(20,28,33,34)(21,36,37,29)(24,31,41,40)(25,42,35,38)(32,46,47,43)(39,45,48,44);(9,10)(11,14)(12,13)(15,16)(17,20,22,33)(18,24,27,41)(19,28,30,34)(21,32,37,47)(23,31,26,40)(25,39,35,48)(29,43,36,46)(38,44,42,45);(3,8)(5,7)(10,13)(14,16)(17,21)(18,25)(19,29)(20,32)(22,37)(23,38)(24,39)(26,42)(27,35)(28,43)(30,36)(31,44)(33,47)(34,46)(40,45)(41,48);(9,11,12,15)(10,14,13,16)(17,19,22,30)(18,26,27,23)(20,28,33,34)(21,36,37,29)(24,40,41,31)(25,38,35,42)(32,46,47,43)(39,44,48,45)

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=i,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, 7077484);