#7011027

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1290 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)(17,18,21,28,22,27,38,26)(19,29,34,45,33,23,32,41)(20,25,37,43,36,44,35,24)(30,40,47,39,31,46,48,42);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,23,27,29)(20,31,36,30)(21,34,38,32)(24,42,43,39)(25,40,44,46)(26,45,28,41)(35,47,37,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,24,27,43)(19,30,33,31)(21,35,38,37)(23,39,29,42)(25,28,44,26)(32,47,34,48)(40,45,46,41);(3,8)(6,7)(11,16)(13,15)(17,20)(18,25)(19,31)(21,37)(22,36)(23,40)(24,26)(27,44)(28,43)(29,46)(30,33)(32,47)(34,48)(35,38)(39,41)(42,45);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,38)(18,26,27,28)(19,32,33,34)(20,35,36,37)(23,41,29,45)(24,25,43,44)(30,47,31,48)(39,40,42,46)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,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, 6918223);