#494491

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1288 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323138
maximal normal subgroup of	#72323138
maximal quotient of	#72323110

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401687);