#148945

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2076
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di,f^b=fh,g^b=gi,h^b=hi,e^c=ei,f^c=fh,g^c=gi,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, 56141);