#495013

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,d^c=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, 402209);