#146807

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

946 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,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, 54003);