#6686204

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	#35697	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 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)(21,22)(23,26)(24,28)(25,27)(29,30)(31,36)(32,35)(33,34);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,32)(22,26,27,35)(24,30,34,31)(28,29,33,36);(1,2,3,4)(21,22)(23,29)(24,33)(25,27)(26,30)(28,34)(31,35)(32,36);(6,11)(10,13)(12,19)(18,20)(22,27)(26,35)(28,33)(29,36);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,30)(24,34)(26,29)(28,33)(31,32)(35,36)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,e^2=f,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, 6593400);