#6441892

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1994 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,31,40)(20,25,34,44)(21,35,22,26)(28,37,30,39)(29,38,45,48)(32,41,46,47)(33,43,36,42);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,32)(21,30)(22,28)(23,31)(25,41)(26,37)(27,40)(29,33)(34,46)(35,39)(36,45)(38,43)(42,48)(44,47);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(19,28)(21,22)(24,37)(26,35)(29,32)(30,31)(33,36)(38,41)(39,40)(42,43)(45,46)(47,48);(1,4,5,3)(2,6,8,7)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,38)(26,42)(27,44)(28,32)(30,46)(31,45)(35,43)(37,41)(39,47)(40,48);(17,21,23,22)(18,26,27,35)(19,28,31,30)(20,33,34,36)(24,39,40,37)(25,42,44,43)(29,32,45,46)(38,47,48,41)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349088);