#148517

order	512
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

634 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)(8,23,27,41)(9,29)(10,34)(11,17)(12,20)(13,26,42,25)(14,32)(15,45)(16,31)(19,50,52,66)(21,46)(22,37)(24,40,51,49)(28,63,60,57)(30,48)(33,47)(35,44)(36,53,55,58)(38,65)(39,54)(43,61,56,62)(59,68,64,67);(1,2)(3,4)(5,7,11,22)(6,12,17,39)(8,24,27,51)(9,30,33,65)(10,35,31,21)(13,40,42,49)(14,44,15,46)(16,48,34,38)(18,47,37,29)(19,25,52,26)(20,45,54,32)(23,55,41,36)(28,64,60,59)(43,68,56,67)(50,57,66,63)(53,62,58,61);(2,4)(5,8,9,25,31,60,45,62,11,27,33,26,10,28,32,61)(6,13,14,23,34,56,47,63,17,42,15,41,16,43,29,57)(7,19,20,51,35,58,65,59,22,52,54,24,21,53,30,64)(12,36,18,49,48,50,46,67,39,55,37,40,38,66,44,68)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=hi,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,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,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55713);