#7162174

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,22,26)(19,27,31,43)(20,32,34,24)(21,36,28,25)(23,37,39,30)(29,45,33,42)(35,41,44,48)(38,47,40,46);(2,5)(3,7)(10,13)(11,15)(17,19,28,37,22,31,21,30)(18,23,36,43,26,39,25,27)(20,33,44,47,34,29,35,46)(24,40,48,42,32,38,41,45);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,34)(18,24,26,32)(19,29,31,33)(21,35,28,44)(23,38,39,40)(25,41,36,48)(27,42,43,45)(30,47,37,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,30)(20,34)(21,28)(23,27)(24,32)(25,36)(29,46)(31,37)(33,47)(38,45)(39,43)(40,42);(9,11,12,15)(10,14,13,16)(19,31)(20,34)(23,39)(24,32)(27,43)(30,37)(35,44)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

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, 7069370);