#149358

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

428 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)(5,6,12,19,22,41,38,54,13,20,9,16,11,18,10,17)(7,21,27,46,26,56,25,14,29,45,24,44,28,43,40,47)(8,30,32,52,34,55,33,51,36,53,39,15,35,49,37,50)(23,48,58,57,62,42,61,66,60,67,65,63,31,64,59,68);(2,4)(5,7,22,28,13,29,11,26)(6,14,41,46,20,47,18,44)(8,31,34,60,36,62,35,23)(9,25,10,27,12,40,38,24)(15,48,50,64,52,67,51,42)(16,43,17,45,19,56,54,21)(30,63,55,66,53,57,49,68)(32,65,33,61,39,58,37,59);(2,4)(5,8,13,36)(6,15,20,52)(7,23)(9,37,12,33)(10,39,38,32)(11,34,22,35)(14,42)(16,53,19,30)(17,55,54,49)(18,50,41,51)(21,57)(24,59)(25,58)(26,31)(27,61)(28,62)(29,60)(40,65)(43,66)(44,48)(45,63)(46,67)(47,64)(56,68)

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=h,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,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, 56554);