#113301

order	512
nilpotency class	3
isoclinism	isoclinic class of #113294
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

757 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:

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

Polycyclic presentation:

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

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, 20497);