#113073

order	512
nilpotency class	3
isoclinism	isoclinic class of #113067
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	#37558	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)(4,11,23,39,21,40,52,63)(6,12,25,41,27,42,24,53)(8,28,32,20,10,37,38,17)(9,33,35,51,36,54,61,22)(18,34,49,60,46,45,64,62)(29,56,57,48,30,55,58,50);(2,8)(3,16)(4,21)(5,24)(9,34)(10,14)(11,29)(12,31)(13,43)(15,42)(17,20)(19,37)(22,51)(23,33)(25,26)(28,47)(30,40)(32,53)(35,58)(36,45)(38,41)(39,62)(44,59)(49,55)(52,54)(56,64)(57,61)(60,63);(2,9)(3,17)(4,22)(5,25)(8,29)(10,30)(11,12)(13,39)(14,36)(15,45)(16,20)(18,46)(19,47)(21,51)(24,26)(28,37)(31,34)(32,60)(35,41)(38,62)(40,42)(43,58)(44,63)(48,50)(49,55)(53,61)(56,64)(57,59)

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,f^a=fh,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=dhi,e^c=eh,g^c=gh

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, 20269);