#148562

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	#1879
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

650 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)(7,20,24,39)(8,26)(9,32)(10,16)(11,36)(12,19)(13,22,41,55)(14,28)(15,33)(17,34)(18,35)(21,51,53,57)(23,49,56,42)(25,40,37,48)(27,59,61,43)(29,46)(30,65)(31,44)(38,54,68,60)(45,64)(47,63)(50,66,67,52)(58,62);(1,2)(3,4)(5,7)(6,13)(8,27)(9,23)(10,22)(11,37)(12,24)(14,43)(15,40)(16,48)(17,20)(18,49)(19,41)(21,29)(25,35)(26,57)(28,60)(30,66)(31,61)(32,39)(33,55)(34,56)(36,42)(38,45)(44,51)(46,67)(47,59)(50,58)(52,64)(53,62)(54,63)(65,68);(2,4)(5,8,9,28,11,30,33,62,12,31,34,63,35,64,10,29)(6,14,15,44,18,46,32,65,19,47,16,26,36,58,17,45)(7,21,22,52,25,54,56,61,24,53,55,66,37,60,23,27)(13,38,20,50,42,57,48,59,41,68,39,67,49,51,40,43)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,f^b=fh,g^b=gi,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, 55758);