#113768

order	512
nilpotency class	3
isoclinism	isoclinic class of #113765
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20964);