#496610

order	512
nilpotency class	4
isoclinism	isoclinic class of #481393
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

758 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=ei,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fhi,g^c=gi,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, 403806);