#494837

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

778 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)(5,6,10,16,11,15,31,34,12,17,32,41,9,30,19,45)(7,18,22,43,23,37,51,60,24,44,52,64,21,13,33,36)(8,25,28,54,29,39,56,61,26,53,55,65,27,14,38,40)(20,46,49,66,50,62,67,68,47,35,58,59,48,42,57,63);(5,7,19,33,9,21,32,52,12,24,31,51,11,23,10,22)(6,13,34,43,30,44,16,36,17,37,45,64,15,18,41,60)(8,20,38,57,27,48,55,58,26,47,56,67,29,50,28,49)(14,35,54,63,53,62,40,59,39,46,65,68,25,42,61,66);(2,4)(5,8,9,27,12,26,11,29)(6,14,15,25,17,39,30,53)(7,20,21,48,24,47,23,50)(10,28,19,38,32,55,31,56)(13,35,18,42,37,46,44,62)(16,40,34,54,41,61,45,65)(22,49,33,57,52,58,51,67)(36,59,43,63,60,66,64,68);(8,26)(14,39)(20,47)(25,53)(27,29)(28,55)(35,46)(38,56)(40,61)(42,62)(48,50)(49,58)(54,65)(57,67)(59,66)(63,68)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,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, 402033);