#6473903

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=f,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^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, 6381099);