#501983

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28245	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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)(3,5)(4,6)(7,8)(9,10)(11,23)(12,18)(13,37)(14,21)(15,19)(16,22)(17,27)(20,24)(25,55)(26,60)(28,43)(29,45)(30,53)(31,64)(32,66)(33,54)(34,49)(35,48)(36,50)(38,51)(39,56)(40,47)(41,52)(42,58)(44,61)(46,62)(57,72)(59,69)(63,70)(65,71)(67,68);(1,3,4,7)(2,5,6,8)(9,11,24,29,16,30,14,27)(10,17,21,53,22,45,20,23)(12,31,35,57,36,65,33,62)(13,26,41,61,38,28,15,39)(18,46,54,71,50,72,48,64)(19,43,51,44,52,60,37,56)(25,40,63,32,59,67,58,34)(42,68,69,66,70,47,55,49);(2,6)(5,8)(9,12)(10,18)(11,25)(13,32)(14,33)(15,40)(16,36)(17,42)(19,47)(20,48)(21,54)(22,50)(23,55)(24,35)(26,57)(27,58)(28,62)(29,63)(30,59)(31,39)(34,38)(37,66)(41,67)(43,46)(44,71)(45,70)(49,51)(52,68)(53,69)(56,64)(60,72)(61,65);(9,13,14,15,16,38,24,41)(10,19,20,37,22,52,21,51)(11,26,27,39,30,28,29,61)(12,32,33,40,36,34,35,67)(17,43,23,56,45,60,53,44)(18,47,48,66,50,68,54,49)(25,57,58,31,59,62,63,65)(42,46,55,64,70,72,69,71)

Polycyclic presentation:

2^9|b^2=efi,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,g^c=gh,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, 409179);