#6460469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1804 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367665);