#502135

order	512
nilpotency class	4
isoclinism	isoclinic class of #502133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,9,15,30,44,33,47,12,18,31,45,11,17,10,16)(7,13,20,36,54,65,21,37,23,39,55,66,22,38,32,46)(8,24,26,48,34,62,61,43,29,59,60,42,28,41,27,14)(19,49,50,68,57,40,25,58,52,67,56,63,51,64,53,35);(2,4)(5,7,9,20,30,54,33,21,12,23,31,55,11,22,10,32)(6,13,15,36,44,65,47,37,18,39,45,66,17,38,16,46)(8,25,26,52,34,56,61,51,29,53,60,19,28,50,27,57)(14,40,24,58,48,67,62,63,43,64,59,35,42,49,41,68);(5,8,12,29)(6,14,18,43)(7,19,23,52)(9,27,31,61)(10,26,33,60)(11,34,30,28)(13,35,39,58)(15,41,45,62)(16,24,47,59)(17,48,44,42)(20,53,55,25)(21,56,32,50)(22,57,54,51)(36,64,66,40)(37,67,46,49)(38,68,65,63);(2,4)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi,h^c=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, 409331);