#148807

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2034
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 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)(8,26)(9,32)(10,36)(11,16)(12,18)(13,21)(14,29)(15,35)(17,34)(20,46)(22,42)(23,43)(24,45)(25,44)(27,62)(28,64)(30,47)(31,52)(33,48)(37,65)(38,50)(39,54)(40,51)(41,60)(49,55)(53,57)(56,67)(58,66)(59,68)(61,63);(5,7,9,21,12,25,35,24)(6,13,15,19,18,45,32,44)(8,27,28,63,31,60,33,57)(10,37,11,39,34,23,40,22)(14,46,47,66,49,68,50,67)(16,43,36,54,51,65,17,42)(20,55,56,30,59,29,58,38)(26,61,48,62,52,53,64,41);(5,8,12,31)(6,14,18,49)(7,20)(9,33,35,28)(10,38,34,30)(11,29,40,55)(13,41)(15,50,32,47)(16,48,51,64)(17,52,36,26)(19,53)(21,58)(22,60)(23,57)(24,56)(25,59)(27,39)(37,63)(42,46)(43,67)(44,61)(45,62)(54,68)(65,66)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fh,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, 56003);