#6441492

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

1882 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,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, 6348688);