#146913

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1824
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,10,17,12,15,11,34)(7,18,21,39,22,13,33,38)(8,23,26,55,24,56,35,45)(9,27,30,43,32,59,31,16)(14,29,40,57,42,63,41,25)(19,47,50,68,48,64,51,66)(20,52,54,44,28,60,62,37)(36,53,65,49,46,61,67,58);(1,2,3,4)(5,7)(6,13)(8,19)(9,28)(10,33)(11,21)(12,22)(14,36)(15,18)(16,44)(17,39)(20,32)(23,47)(24,48)(25,58)(26,51)(27,52)(29,61)(30,54)(31,62)(34,38)(35,50)(37,43)(40,67)(41,65)(42,46)(45,68)(49,57)(53,63)(55,66)(56,64)(59,60);(5,8,11,35,12,24,10,26)(6,14,34,41,15,42,17,40)(7,19,21,50,22,48,33,51)(9,29,31,25,32,63,30,57)(13,36,38,65,18,46,39,67)(16,45,59,56,43,55,27,23)(20,53,54,49,28,61,62,58)(37,66,52,47,44,68,60,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,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, 54109);