#494838

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

782 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)(7,18,23,44,24,36,52,60,21,43,51,64,22,13,33,37)(8,25,28,54,29,39,56,61,26,53,55,65,27,14,38,40)(20,46,48,66,49,63,67,68,50,35,58,59,47,42,57,62);(5,7,19,33,9,22,32,51,12,21,31,52,11,24,10,23)(6,13,34,44,30,43,16,37,17,36,45,64,15,18,41,60)(8,20,38,57,27,47,55,58,26,50,56,67,29,49,28,48)(14,35,54,62,53,63,40,59,39,46,65,68,25,42,61,66);(2,4)(5,8,9,27,12,26,11,29)(6,14,15,25,17,39,30,53)(7,20,22,47,21,50,24,49)(10,28,19,38,32,55,31,56)(13,35,18,42,36,46,43,63)(16,40,34,54,41,61,45,65)(23,48,33,57,51,58,52,67)(37,59,44,62,60,66,64,68);(7,21)(8,26)(13,36)(14,39)(18,43)(22,24)(23,51)(25,53)(27,29)(28,55)(33,52)(37,60)(38,56)(40,61)(44,64)(54,65)

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^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, 402034);