#498518

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

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	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,20,15,19,34,43,16,21,35,44,13,32,33,58)(11,17,24,38,25,37,52,62,26,39,53,63,23,50,51,69)(12,27,30,55,31,41,57,64,28,54,56,70,29,18,40,42)(22,45,48,66,49,60,68,71,46,65,67,72,47,36,59,61);(2,5)(3,7)(9,11,14,24,15,25,34,52,16,26,35,53,13,23,33,51)(10,17,20,38,19,37,43,62,21,39,44,63,32,50,58,69)(12,22,30,48,31,49,57,68,28,46,56,67,29,47,40,59)(18,36,42,61,27,45,55,66,41,60,64,71,54,65,70,72);(9,12,13,29,16,28,15,31)(10,18,19,27,21,41,32,54)(11,22,23,47,26,46,25,49)(14,30,33,40,35,56,34,57)(17,36,37,45,39,60,50,65)(20,42,43,55,44,64,58,70)(24,48,51,59,53,67,52,68)(38,61,62,66,63,71,69,72);(1,3,4,7)(2,6,5,8)(12,28)(18,41)(22,46)(27,54)(29,31)(30,56)(36,60)(40,57)(42,64)(45,65)(47,49)(48,67)(55,70)(59,68)(61,71)(66,72)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,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, 405714);