#496032

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

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	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

522 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)(9,10,15,22,16,21,35,51,17,23,34,19,14,41,12,31)(11,24,28,52,29,44,25,45,30,54,56,60,27,18,32,42)(13,20,38,49,39,48,64,70,40,50,63,46,37,67,33,61)(26,53,58,71,59,47,55,69,36,65,66,72,57,43,62,68);(2,5)(6,8)(9,11,14,27,17,30,16,29)(10,18,21,24,23,44,41,54)(12,32,34,56,35,25,15,28)(13,36,37,59,40,26,39,57)(19,45,31,60,22,42,51,52)(20,47,48,65,50,43,67,53)(33,55,63,58,64,62,38,66)(46,68,61,71,49,69,70,72);(1,3,4,7)(2,6,5,8)(9,12,14,34,17,35,16,15)(10,19,21,31,23,22,41,51)(11,25,27,28,30,32,29,56)(13,33,37,63,40,64,39,38)(18,42,24,52,44,45,54,60)(20,46,48,61,50,49,67,70)(26,55,57,58,36,62,59,66)(43,68,53,71,47,69,65,72);(9,13,15,38,16,39,35,64,17,40,34,63,14,37,12,33)(10,20,22,49,21,48,51,70,23,50,19,46,41,67,31,61)(11,26,28,58,29,59,25,55,30,36,56,66,27,57,32,62)(18,43,42,68,24,53,52,71,44,47,45,69,54,65,60,72)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=g,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=ci,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403228);