#354644

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1566 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)(5,6,9,13,10,14,21,27)(7,15,18,29,17,11,22,24)(8,12,20,26,19,25,32,35)(16,28,31,36,30,23,33,34);(5,7)(6,11,14,15)(8,16)(9,18)(10,17)(12,23,25,28)(13,24,27,29)(19,30)(20,31)(21,22)(26,34,35,36)(32,33);(1,2,3,4)(8,19)(12,25)(16,30)(20,32)(23,28)(26,35)(31,33)(34,36);(5,8,9,20,10,19,21,32)(6,12,13,26,14,25,27,35)(7,16,18,31,17,30,22,33)(11,23,24,34,15,28,29,36)

Polycyclic presentation:

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

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, 261840);