#6676689

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1855 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)(7,10)(8,12)(9,11)(14,19)(16,28)(20,33)(22,23)(24,39)(25,29)(30,42)(41,44);(5,7,10,6)(8,12,9,11)(13,14)(15,24)(16,21)(17,19)(18,22)(20,27)(23,31)(25,40)(26,39)(28,34)(29,43)(30,37)(32,33)(35,41)(36,44)(38,42);(2,4)(13,15,17,26)(14,20,19,33)(16,29,28,25)(18,32,31,27)(21,36,34,35)(22,24,23,39)(30,41,42,44)(37,43,38,40);(5,8,10,9)(6,11,7,12)(13,16,17,28)(14,21,19,34)(15,25,26,29)(18,30,31,42)(20,35,33,36)(22,37,23,38)(24,40,39,43)(27,41,32,44);(5,9,10,8)(6,11,7,12)(14,22)(18,31)(19,23)(20,24)(21,37)(27,32)(30,42)(33,39)(34,38)(35,40)(36,43)(41,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,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, 6583885);