#496145

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
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×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

476 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)(3,8)(4,6)(5,7)(9,10,15,22,16,21,34,51,17,23,33,19,14,40,12,30)(11,18,27,41,28,43,24,52,29,44,54,45,26,58,31,59)(13,20,37,49,38,48,63,70,39,50,62,46,36,65,32,60)(25,42,56,66,57,67,53,71,35,47,64,68,55,69,61,72);(1,3,5,4)(2,6,7,8)(9,11)(10,18)(12,31)(13,35)(14,26)(15,27)(16,28)(17,29)(19,45)(20,47)(21,43)(22,41)(23,44)(24,34)(25,39)(30,59)(32,53)(33,54)(36,57)(37,64)(38,55)(40,58)(42,50)(46,66)(48,69)(49,68)(51,52)(56,62)(60,71)(61,63)(65,67)(70,72);(9,12,14,33,17,34,16,15)(10,19,21,30,23,22,40,51)(11,24,26,27,29,31,28,54)(13,32,36,62,39,63,38,37)(18,41,43,52,44,45,58,59)(20,46,48,60,50,49,65,70)(25,53,55,56,35,61,57,64)(42,66,67,71,47,68,69,72);(1,4,5,3)(2,6,7,8)(9,13,15,37,16,38,34,63,17,39,33,62,14,36,12,32)(10,20,22,49,21,48,51,70,23,50,19,46,40,65,30,60)(11,25,27,56,28,57,24,53,29,35,54,64,26,55,31,61)(18,42,41,66,43,67,52,71,44,47,45,68,58,69,59,72)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=eg,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,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, 403341);