#7074157

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1296 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=hi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,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, 6981353);