#149271

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56467);