#113409

order	512
nilpotency class	3
isoclinism	isoclinic class of #113404
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=di,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, 20605);