#146912

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	#1804
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

494 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)(7,18,21,13)(8,24)(9,28)(10,17)(11,34)(12,15)(14,26)(16,32)(19,47,48,64)(20,51,29,61)(22,46,23,38)(25,57)(27,56)(30,39)(31,42)(33,60)(35,44)(36,49,45,59)(37,54,43,53)(40,63)(41,58)(50,67,55,68)(52,66,62,65);(1,2,3,4)(5,7,10,22,12,21,11,23)(6,13,17,38,15,18,34,46)(8,19,27,50,25,48,35,55)(9,29,31,54,33,20,32,53)(14,36,39,65,41,45,40,66)(16,43,28,51,42,37,60,61)(24,47,56,67,57,64,44,68)(26,59,30,62,58,49,63,52);(5,8,11,35,12,25,10,27)(6,14,34,40,15,41,17,39)(7,19,23,55,21,48,22,50)(9,30,32,26,33,63,31,58)(13,36,46,66,18,45,38,65)(16,44,60,57,42,56,28,24)(20,52,54,49,29,62,53,59)(37,67,51,47,43,68,61,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,e^2=hi,g^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, 54108);