#148920

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

458 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,g^b=gi,e^c=ei,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, 56116);