#123873

order	512
nilpotency class	3
isoclinism	isoclinic class of #9109
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1598 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:

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

Polycyclic presentation:

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

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, 31069);