#494930

order	512
nilpotency class	4
isoclinism	isoclinic class of #483839
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
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

750 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)(3,4)(5,6,9,16,13,19,12,38)(7,20,22,52,25,42,33,53)(8,26,31,62,32,48,28,61)(10,36,34,18,11,37,35,17)(14,39,41,23,44,59,43,24)(15,45,49,30,50,66,47,29)(21,54,57,40,58,46,27,63)(51,65,60,56,67,64,68,55);(5,7,10,24,12,22,35,39,13,25,11,23,9,33,34,59)(6,14,18,20,16,43,37,53,19,44,17,42,38,41,36,52)(8,27,30,64,31,58,66,56,32,57,29,65,28,21,45,55)(15,46,48,68,47,63,62,51,50,54,26,60,49,40,61,67);(2,4)(5,8,10,29,12,31,35,45,13,32,11,30,9,28,34,66)(6,15,17,48,16,47,36,62,19,50,18,26,38,49,37,61)(7,21,23,55,22,27,59,64,25,58,24,56,33,57,39,65)(14,40,20,51,43,46,53,60,44,63,42,67,41,54,52,68);(5,9,13,12)(6,16,19,38)(7,22,25,33)(8,28,32,31)(10,34,11,35)(14,41,44,43)(15,47,50,49)(17,36,18,37)(20,52,42,53)(21,27,58,57)(23,59,24,39)(26,61,48,62)(29,66,30,45)(40,54,63,46)(51,68,67,60)(55,64,56,65)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343347, #72343350, #72343366, …
maximal normal subgroup of	#72343347, #72343350, #72343366, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402126);