#497274

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

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	C₂²×C₂²⋊C₈
prime radical	D₄×#41	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404470);