#123939

order	512
nilpotency class	3
isoclinism	isoclinic class of #2008
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1444
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31135);