#499991

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61261	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

892 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)(11,17)(12,30)(13,19)(14,39)(15,20)(16,23)(18,33)(21,37)(22,38)(24,57)(25,42)(26,52)(27,43)(28,56)(29,45)(31,64)(32,48)(34,47)(35,55)(36,53)(40,51)(41,59)(44,63)(46,60)(49,68)(50,67)(54,62)(58,71)(61,72)(65,70)(66,69);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,31)(13,25)(14,27)(15,26)(16,29)(18,46)(19,42)(20,52)(21,54)(22,56)(23,45)(24,34)(28,38)(30,64)(32,59)(33,60)(35,66)(36,65)(37,62)(39,43)(40,63)(41,48)(44,51)(47,57)(49,72)(50,71)(53,70)(55,69)(58,67)(61,68);(9,12,16,35)(10,18,23,50)(11,24,29,61)(13,36,38,32)(14,34,37,68)(15,33,40,67)(17,41,45,70)(19,49,22,47)(20,53,51,48)(21,55,39,30)(25,60,28,58)(26,31,63,66)(27,59,62,65)(42,64,56,69)(43,71,54,46)(44,72,52,57);(2,6)(5,7)(9,13,16,38)(10,19,23,22)(11,25,29,28)(12,32,35,36)(14,15,37,40)(17,42,45,56)(18,47,50,49)(20,21,51,39)(24,58,61,60)(26,62,63,27)(30,48,55,53)(31,59,66,65)(33,34,67,68)(41,69,70,64)(43,52,54,44)(46,57,71,72)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,d^c=di,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407187);