#7197677

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 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)(10,11)(12,16)(17,18,22,27,23,28,39,46)(19,29,31,41,32,45,36,24)(20,33,35,48,34,25,42,40)(21,26,38,44,37,43,30,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19,23,32)(18,24,28,41)(20,30,34,38)(21,35,37,42)(22,36,39,31)(25,26,33,43)(27,45,46,29)(40,47,48,44);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,20,23,34)(18,25,28,33)(19,30,32,38)(21,36,37,31)(22,35,39,42)(24,26,41,43)(27,40,46,48)(29,44,45,47);(3,8)(5,7)(10,11)(14,15)(17,21,22,38,23,37,39,30)(18,26,27,44,28,43,46,47)(19,20,31,35,32,34,36,42)(24,40,29,33,41,48,45,25);(9,12,13,16)(10,14,11,15)(20,34)(21,37)(25,33)(26,43)(30,38)(35,42)(40,48)(44,47)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=hi,d^2=fgi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7104873);