#493182

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,d^a=di,g^a=gi,d^b=dh,e^b=ei,d^c=di,g^d=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, 400378);