#149298

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56494);