#149146

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

420 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gi,h^b=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, 56342);