#6473658

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,28)(19,26,31,24)(20,32,34,47)(21,27,23,37)(25,33,42,35)(29,43,45,46)(30,40,36,39)(38,44,41,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,36)(22,31)(23,30)(25,38)(26,28)(27,40)(32,46)(33,48)(34,45)(35,44)(37,39)(41,42)(43,47);(11,15)(13,16)(17,20,23,35,22,34,21,33)(18,25,27,32,28,42,37,47)(19,29,30,44,31,45,36,48)(24,38,40,46,26,41,39,43);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,19)(18,26)(20,29)(21,36)(22,31)(23,30)(24,28)(25,41)(27,39)(32,43)(33,48)(34,45)(35,44)(37,40)(38,42)(46,47);(10,14)(13,16)(17,21,22,23)(18,27,28,37)(19,30,31,36)(20,33,34,35)(24,39,26,40)(25,32,42,47)(29,44,45,48)(38,43,41,46)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=fi,e^2=i,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^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6380854);