#149409

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1970
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=deh,d^2=hi,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=ghi,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, 56605);