#148934

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	#1832
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

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

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, 56130);