#149220

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2085
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=eh,d^2=h,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 56416);