#113082

order	512
nilpotency class	3
isoclinism	isoclinic class of #113076
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	#37543	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

806 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:

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

Polycyclic presentation:

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

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, 20278);