#146914

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,e^2=hi,f^2=i,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, 54110);