#146805

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,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, 54001);