#502339

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

501 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,7)(4,5)(6,8)(9,10,13,19,34,48,37,51,16,22,35,49,15,21,14,20)(11,17,24,40,58,69,25,41,27,43,59,70,26,42,36,50)(12,28,30,52,38,66,65,47,33,63,64,46,32,45,31,18)(23,53,54,72,61,44,29,62,56,71,60,67,55,68,57,39);(2,5)(6,7)(9,11,13,24,34,58,37,25,16,27,35,59,15,26,14,36)(10,17,19,40,48,69,51,41,22,43,49,70,21,42,20,50)(12,29,30,56,38,60,65,55,33,57,64,23,32,54,31,61)(18,44,28,62,52,71,66,67,47,68,63,39,46,53,45,72);(9,12,16,33)(10,18,22,47)(11,23,27,56)(13,31,35,65)(14,30,37,64)(15,38,34,32)(17,39,43,62)(19,45,49,66)(20,28,51,63)(21,52,48,46)(24,57,59,29)(25,60,36,54)(26,61,58,55)(40,68,70,44)(41,71,50,53)(42,72,69,67);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=e,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, 409535);