#6676672

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#42800	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1856 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)(13,14)(15,20)(16,21)(17,22)(18,24)(19,23)(25,36)(26,28)(27,40)(29,37)(30,38)(31,35)(32,34)(33,39)(41,44)(42,43);(5,7,10,6)(8,12,9,11)(13,15)(14,20)(16,30)(17,26)(18,27)(19,29)(21,38)(22,28)(23,37)(24,40)(25,32)(31,44)(33,43)(34,36)(35,41)(39,42);(2,4)(13,16,19,33)(14,21,23,39)(15,25,29,41)(17,34,24,31)(18,35,22,32)(20,36,37,44)(26,42,40,38)(27,30,28,43);(5,8,10,9)(6,11,7,12)(13,17,19,24)(14,22,23,18)(15,26,29,40)(16,31,33,34)(20,28,37,27)(21,35,39,32)(25,38,41,42)(30,44,43,36);(5,9,10,8)(6,11,7,12)(14,23)(15,27)(18,22)(20,26)(21,39)(25,30)(28,29)(32,35)(36,42)(37,40)(38,44)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583868);