#495012

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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 ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

668 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)(7,13,20,34,21,33,48,58,22,35,49,59,19,46,47,65)(8,23,26,51,27,37,53,60,24,50,52,66,25,14,36,38)(18,41,44,62,45,56,64,67,42,61,63,68,43,32,55,57);(2,4)(5,7,10,20,11,21,30,48,12,22,31,49,9,19,29,47)(6,13,16,34,15,33,39,58,17,35,40,59,28,46,54,65)(8,18,26,44,27,45,53,64,24,42,52,63,25,43,36,55)(14,32,38,57,23,41,51,62,37,56,60,67,50,61,66,68);(5,8,9,25,12,24,11,27)(6,14,15,23,17,37,28,50)(7,18,19,43,22,42,21,45)(10,26,29,36,31,52,30,53)(13,32,33,41,35,56,46,61)(16,38,39,51,40,60,54,66)(20,44,47,55,49,63,48,64)(34,57,58,62,59,67,65,68);(8,24)(14,37)(18,42)(23,50)(25,27)(26,52)(32,56)(36,53)(38,60)(41,61)(43,45)(44,63)(51,66)(55,64)(57,67)(62,68)

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^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, 402208);