#148456

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

648 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)(5,6,11,16,9,17,8,24,12,18,26,14,10,15,27,34)(7,19,20,29,22,32,25,33,23,30,35,36,21,13,28,31);(5,7)(6,13)(8,25)(9,22)(10,21)(11,20)(12,23)(14,33)(15,30)(16,31)(17,19)(18,32)(24,29)(26,35)(27,28)(34,36);(2,4)(5,8,9,26,12,27,10,11)(6,14,15,24,18,16,17,34)(7,20,21,28,23,35,22,25)(13,29,19,33,32,36,30,31)

Polycyclic presentation:

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

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, 55652);