#6474555

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1319 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6381751);