#148204

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1804
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

678 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,e^b=ei,d^c=di,e^c=eh,f^c=fi,g^c=gh,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, 55400);