#6441495

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=i,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, 6348691);