#149096

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	#2101
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

450 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=i,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fhi,g^b=gh,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, 56292);