#6447107

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fgi,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, 6354303);