#6716797

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,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, 6623993);