#6494326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2476 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6401522);