#148366

order	512
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2065
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 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)(5,6,12,18,10,16,13,29)(7,19,23,31,24,36,28,35)(8,9,26,27,25,15,17,11)(14,30,32,21,33,22,20,34);(1,2,3,4)(5,7,9,21,13,23,11,22,10,24,15,34,12,28,27,30)(6,14,8,19,18,20,26,35,16,33,25,36,29,32,17,31);(5,8)(6,15)(7,20)(9,16)(10,25)(11,18)(12,26)(13,17)(14,23)(19,34)(21,36)(22,35)(24,32)(27,29)(28,33)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55562);