#6683737

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2273 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,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, 6590933);