#148159

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1862
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,d^c=di,e^c=eh,f^c=fh,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, 55355);