#498848

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1534 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)(3,4)(9,10,15,20)(11,21,23,53)(12,26,29,61)(13,30,35,60)(14,36,33,19)(16,39,41,24)(17,43,45,27)(18,46,49,34)(22,40,54,66)(25,58,37,47)(28,62,38,48)(31,50,63,42)(32,51,64,44)(52,70,71,55)(56,69,59,68)(57,67,65,72);(2,4)(5,6)(7,8)(9,11)(10,16)(12,22)(13,31)(14,37)(15,23)(17,40)(18,47)(19,50)(20,41)(21,30)(24,34)(25,33)(26,52)(27,57)(28,59)(29,54)(32,55)(35,63)(36,42)(38,56)(39,46)(43,65)(44,69)(45,66)(48,67)(49,58)(51,68)(53,60)(61,71)(62,72)(64,70);(6,8)(9,12,15,29)(10,17,20,45)(11,22,23,54)(13,32,35,64)(14,38,33,28)(16,40,41,66)(18,48,49,62)(19,51,36,44)(21,52,53,71)(24,57,39,65)(25,59,37,56)(26,60,61,30)(27,46,43,34)(31,55,63,70)(42,69,50,68)(47,67,58,72);(11,23)(16,41)(21,53)(22,54)(24,39)(25,37)(31,63)(40,66)(42,50)(47,58)(52,71)(55,70)(56,59)(57,65)(67,72)(68,69)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406044);