#503440

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61090	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=efh,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=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, 410636);