#494492

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	C₂²×#508	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)(5,6,10,17,11,18,33,44,12,15,34,45,32,46,9,16)(7,19,23,40,24,47,22,49,21,52,30,13,35,38,57,39)(8,25,27,41,28,48,26,50,29,60,31,14,36,42,61,43)(20,37,55,51,56,63,58,65,53,64,68,67,59,66,54,62);(1,2,3,4)(5,7,11,35,12,21,32,24)(6,13,18,49,15,40,46,39)(8,20,36,56,29,53,28,59)(9,30,10,22,33,23,34,57)(14,37,50,63,41,64,43,66)(16,47,17,19,44,38,45,52)(25,51,42,65,60,67,48,62)(26,55,27,58,61,68,31,54);(5,8)(6,14)(7,20)(9,31)(10,26)(11,36)(12,29)(13,37)(15,41)(16,48)(17,25)(18,50)(19,51)(21,53)(22,55)(23,58)(24,59)(27,33)(28,32)(30,54)(34,61)(35,56)(38,65)(39,66)(40,64)(42,44)(43,46)(45,60)(47,62)(49,63)(52,67)(57,68);(6,15)(7,21)(14,41)(16,44)(17,45)(18,46)(20,53)(22,57)(23,30)(24,35)(25,60)(42,48)(43,50)(54,58)(55,68)(56,59)

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^a=di,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	#72323048

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401688);