#494414

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

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×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,10,17,11,18,30,21,12,15,31,35,9,16,20,33)(7,19,14,34,23,8,24,26,22,25,28,36,27,13,32,29);(1,2,3,4)(5,7,20,24,9,27,31,14,12,22,30,32,11,23,10,28)(6,13,33,34,16,25,35,29,15,8,21,36,18,19,17,26);(5,8)(6,14)(7,21)(9,19)(10,29)(11,25)(12,13)(15,28)(16,32)(17,27)(18,24)(20,36)(22,33)(23,35)(26,31)(30,34);(6,15)(7,22)(14,28)(16,18)(17,35)(21,33)(23,27)(24,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

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, 401610);