#506195

order	512
nilpotency class	4
isoclinism	isoclinic class of #500843
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	#24280	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

655 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)(11,16)(12,26)(13,18)(14,20)(15,21)(17,29)(19,33)(22,49)(23,37)(24,39)(25,40)(27,57)(28,44)(30,58)(31,43)(32,45)(34,42)(35,48)(36,51)(38,54)(41,55)(46,63)(47,62)(50,65)(52,64)(53,66)(56,69)(59,70)(60,68)(61,71)(67,72);(1,3,4,8)(2,5,6,7)(9,11,14,24,33,54,63,72,15,25,35,56,13,23,32,53)(10,16,20,39,19,38,46,67,21,40,48,69,18,37,45,66)(12,27,30,50,28,59,61,52,31,60,62,51,29,55,34,22)(17,41,42,49,26,57,58,65,44,70,71,64,43,68,47,36);(9,12)(10,17)(11,22)(13,28)(14,34)(15,31)(16,36)(18,26)(19,43)(20,47)(21,44)(23,50)(24,55)(25,52)(27,53)(29,33)(30,32)(35,61)(37,49)(38,64)(39,68)(40,65)(41,66)(42,45)(46,71)(48,58)(51,54)(56,59)(57,69)(60,72)(62,63)(67,70);(2,6)(5,7)(9,13,15,33)(10,18,21,19)(11,23,25,54)(12,28,31,29)(14,32,35,63)(16,37,40,38)(17,26,44,43)(20,45,48,46)(22,50,52,51)(24,53,56,72)(27,59,60,55)(30,61,62,34)(36,49,65,64)(39,66,69,67)(41,57,70,68)(42,58,71,47)

Polycyclic presentation:

2^9|b^2=eg,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413391);