#353328

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3982 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)(3,4)(5,6,9,16,11,15,10,14)(7,17,20,12,19,28,21,27)(8,22,24,30,25,13,23,29)(18,32,35,26,33,31,34,36);(2,4)(5,7,10,21,11,19,9,20)(6,12,16,28,15,27,14,17)(8,18,24,35,25,33,23,34)(13,26,30,32,22,36,29,31);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(27,36)(28,32);(7,19)(12,27)(17,28)(18,33)(20,21)(26,36)(31,32)(34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,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, 260524);