#498549

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

652 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)(5,6,11,18,12,19,34,49,13,20,33,48,10,17,9,16)(7,21,23,46,24,40,31,51,25,50,36,61,35,14,37,39)(8,15,28,43,29,44,26,41,30,45,58,66,27,42,32,47)(22,52,55,68,54,64,60,62,56,67,53,65,59,38,57,63);(5,7,9,31,10,35,33,23,13,25,34,37,12,24,11,36)(6,14,16,46,17,50,48,39,20,40,49,61,19,21,18,51)(8,22,32,60,27,59,58,55,30,56,26,57,29,54,28,53)(15,38,47,68,42,67,66,63,45,64,41,65,44,52,43,62);(1,2,3,4)(5,8)(6,15)(7,22)(9,32)(10,27)(11,28)(12,29)(13,30)(14,38)(16,47)(17,42)(18,43)(19,44)(20,45)(21,52)(23,55)(24,54)(25,56)(26,34)(31,60)(33,58)(35,59)(36,53)(37,57)(39,63)(40,64)(41,49)(46,68)(48,66)(50,67)(51,62)(61,65);(5,9,10,33,13,34,12,11)(6,16,17,48,20,49,19,18)(7,23,24,31,25,36,35,37)(8,26,27,28,30,32,29,58)(14,39,21,46,40,51,50,61)(15,41,42,43,45,47,44,66)(22,53,54,57,56,55,59,60)(38,62,52,65,64,63,67,68)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405745);