#6668233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2670 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

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, 6575429);