#6694309

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1193 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=f,e^2=fi,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, 6601505);