#123753

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1443
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1474 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)(7,17,21,31)(8,22)(9,14)(10,15)(11,16)(12,19,28,20)(13,25)(18,27,23,35)(24,29)(26,30)(32,34,36,33);(5,7,10,20,11,21,9,19)(6,12,14,17,16,28,15,31)(8,23,24,34,26,18,25,33)(13,27,30,36,29,35,22,32);(2,4)(5,8,9,24,11,26,10,25)(6,13,14,22,16,29,15,30)(7,18,19,33,21,23,20,34)(12,27,17,32,28,35,31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,e^c=ei,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, 30949);