#148801

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2034
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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)(7,18)(8,24)(9,29)(10,34)(11,37)(12,17)(13,20)(14,27)(15,33)(16,31)(19,55)(21,41)(22,44)(23,43)(25,62)(26,64)(28,47)(30,51)(32,52)(35,54)(36,49)(38,50)(39,46)(40,60)(42,61)(45,58)(48,65)(53,57)(56,68)(59,67)(63,66);(5,7,9,20,12,23,33,22)(6,13,15,18,17,44,29,43)(8,25,26,60,28,57,30,63)(10,35,31,21,11,38,32,61)(14,45,46,67,48,68,49,55)(16,50,34,41,52,54,37,42)(19,39,56,27,59,36,58,65)(24,40,51,62,47,66,64,53);(5,8,12,28)(6,14,17,48)(7,19)(9,30,33,26)(10,36,11,39)(13,40)(15,49,29,46)(16,51,52,64)(18,53)(20,58)(21,60)(22,56)(23,59)(24,37,47,34)(25,38)(27,32,65,31)(35,57)(41,55)(42,67)(43,62)(44,66)(45,50)(54,68)(61,63)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,d^c=di,e^c=eh,f^c=fhi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55997);