#6444893

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2023 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,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, 6352089);