#146804

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	#1786
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

966 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)(7,18,22,52)(8,24)(9,29)(10,34)(11,17)(12,20,39,21)(13,26)(14,32)(15,45)(16,37)(19,38,54,66)(23,43,35,46)(25,42)(27,47)(28,61)(30,49,57,40)(31,41)(33,48)(36,44)(50,62)(51,63,60,53)(55,68,59,67)(56,64,58,65);(1,2,3,4)(5,7)(6,12)(8,19)(9,30)(10,35)(11,22)(13,38)(14,43)(15,46)(16,49)(17,39)(18,48)(20,45)(21,32)(23,37)(24,60)(25,58)(26,56)(27,59)(28,54)(29,52)(31,63)(33,57)(34,40)(36,55)(41,68)(42,66)(44,65)(47,64)(50,67)(51,61)(53,62);(5,8)(6,13)(7,19)(9,31)(10,36)(11,28)(12,38)(14,44)(15,47)(16,50)(17,42)(18,51)(20,56)(21,58)(22,54)(23,59)(24,29)(25,32)(26,45)(27,37)(30,63)(33,62)(34,41)(35,55)(39,66)(40,68)(43,65)(46,64)(48,61)(49,67)(52,60)(53,57)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,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, 54000);