#7170256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1427 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)(20,31,34,40,33,39,28,24)(21,25,29,41,36,42,35,37)(32,44,45,48,46,47,43,38);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,22,30)(18,23,27,26)(20,28,33,34)(21,35,36,29)(24,31,40,39)(25,41,42,37)(32,45,46,43)(38,47,48,44);(9,10)(11,14)(12,13)(15,16)(17,20,22,33)(18,24,27,40)(19,28,30,34)(21,32,36,46)(23,31,26,39)(25,38,42,48)(29,43,35,45)(37,44,41,47);(3,8)(5,7)(10,13)(14,16)(17,21)(18,25)(19,29)(20,32)(22,36)(23,37)(24,38)(26,41)(27,42)(28,43)(30,35)(31,44)(33,46)(34,45)(39,47)(40,48);(9,11,12,15)(10,14,13,16)(17,19,22,30)(18,26,27,23)(20,28,33,34)(21,35,36,29)(24,39,40,31)(25,37,42,41)(32,45,46,43)(38,44,48,47)

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,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7077452);