#6686203

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	#42971	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,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, 6593399);