#6441952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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

1918 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fi,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,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, 6349148);