#146754

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

960 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)(7,19)(8,25,28,45)(9,30)(10,15)(11,35)(12,18)(13,21)(14,27,43,64)(16,33)(17,32)(20,52,55,67)(22,40)(23,60)(24,47)(26,50,63,44)(29,48,37,42)(31,49)(34,46)(36,39)(38,61,66,56)(41,58)(51,54,68,57)(53,62,59,65);(1,2,3,4)(5,7)(6,13)(8,20)(9,31)(10,21)(11,36)(12,24)(14,38)(15,39)(16,47)(17,49)(18,41)(19,30)(22,33)(23,32)(25,62)(26,57)(27,53)(28,55)(29,61)(34,58)(35,40)(37,56)(42,51)(43,66)(44,67)(45,65)(46,60)(48,68)(50,52)(54,63)(59,64);(5,8)(6,14)(7,20)(9,27)(10,26)(11,37)(12,28)(13,38)(15,45)(16,48)(17,50)(18,43)(19,51)(21,57)(22,59)(23,61)(24,55)(25,46)(29,32)(30,42)(31,53)(33,64)(34,63)(35,44)(36,56)(39,65)(40,67)(41,66)(47,68)(49,52)(54,58)(60,62)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dh,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53950);