#148498

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1934
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

646 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)(5,6,10,16,12,17,11,35)(7,18,21,49,22,50,36,46)(8,23,27,57,29,43,28,61)(9,30,32,47,34,63,33,15)(13,31,38,55,40,64,39,20)(14,41,44,25,45,60,56,26)(19,51,53,37,54,65,58,66)(24,48,59,68,52,42,62,67);(5,7)(6,13)(8,24)(9,31)(10,21)(11,36)(12,22)(14,42)(15,46)(16,38)(17,40)(18,30)(19,26)(20,33)(23,37)(25,54)(27,59)(28,62)(29,52)(32,55)(34,64)(35,39)(41,53)(43,66)(44,67)(45,48)(47,49)(50,63)(51,61)(56,68)(57,65)(58,60);(2,4)(5,8,9,25,10,27,32,60,12,29,34,26,11,28,33,41)(6,14,15,43,35,56,63,57,17,45,47,23,16,44,30,61)(7,19,20,52,36,58,64,59,22,54,55,24,21,53,31,62)(13,37,18,48,38,65,49,68,40,66,50,42,39,51,46,67)

Polycyclic presentation:

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

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, 55694);