#495803

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	#3054	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

652 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)(5,6,10,16,11,15,31,41,12,17,32,42,9,29,30,57)(7,18,22,45,23,34,50,59,19,43,46,65,21,13,33,36)(8,14,27,40,28,39,56,64,25,38,53,63,26,54,55,68)(20,44,48,66,49,37,61,62,24,51,52,67,47,35,58,60);(5,7,9,21,12,19,11,23)(6,13,15,18,17,34,29,43)(8,24,26,49,25,20,28,47)(10,22,30,33,32,46,31,50)(14,37,39,51,38,35,54,44)(16,36,41,45,42,59,57,65)(27,52,55,61,53,48,56,58)(40,62,64,67,63,60,68,66);(2,4)(7,19)(8,25)(13,34)(14,38)(18,43)(21,23)(22,46)(26,28)(27,53)(33,50)(36,59)(39,54)(40,63)(45,65)(55,56)(64,68);(5,8,10,27,11,28,31,56,12,25,32,53,9,26,30,55)(6,14,16,40,15,39,41,64,17,38,42,63,29,54,57,68)(7,20,22,48,23,49,50,61,19,24,46,52,21,47,33,58)(13,35,36,60,18,44,45,66,34,37,59,62,43,51,65,67)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=ci,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, 402999);