#503433

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
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	#61086	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410629);