#7188558

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1233 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)(4,8)(5,7)(10,14)(12,15)(17,18,22,25)(19,27,30,42)(20,32,34,24)(21,36,37,26)(23,29,39,31)(28,44,33,43)(35,41,48,47)(38,45,40,46);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,12,16,15)(17,19,22,30)(18,23,25,39)(20,33,34,28)(21,29,37,31)(24,40,32,38)(26,42,36,27)(35,45,48,46)(41,43,47,44);(2,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,34)(18,24,25,32)(19,28,30,33)(21,35,37,48)(23,38,39,40)(26,41,36,47)(27,43,42,44)(29,46,31,45);(1,4,5,3)(2,8,7,6)(10,14)(11,16)(18,25)(19,29)(20,34)(21,37)(23,27)(28,45)(30,31)(33,46)(38,44)(39,42)(40,43)(41,47);(9,12,13,15)(10,11,14,16)(19,30)(20,34)(23,39)(24,32)(27,42)(29,31)(35,48)(41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7095754);