#148493

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#2033
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

640 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)(5,6,11,18,13,19,12,38)(7,20,24,54,25,52,39,41)(8,26,31,47,32,62,37,46)(9,33,35,51,36,17,10,16)(14,34,42,60,44,23,43,22)(15,28,48,29,50,30,49,65)(21,40,57,67,59,68,58,61)(27,63,64,53,55,66,56,45);(5,7)(6,14)(8,27)(9,34)(10,22)(11,24)(12,39)(13,25)(15,45)(16,41)(17,52)(18,42)(19,44)(20,33)(21,30)(23,36)(26,61)(28,59)(29,58)(31,64)(32,55)(35,60)(37,56)(38,43)(40,47)(46,68)(48,63)(49,66)(50,53)(51,54)(57,65)(62,67);(2,4)(5,8,9,28,11,37,35,65,13,32,36,30,12,31,10,29)(6,15,16,46,38,48,17,26,19,50,51,47,18,49,33,62)(7,21,22,55,39,57,23,56,25,59,60,27,24,58,34,64)(14,40,20,53,42,61,54,63,44,68,52,45,43,67,41,66)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,h^b=hi,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, 55689);