#494301

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,h^a=hi,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fhi,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, 401497);