#114721

order	512
nilpotency class	3
isoclinism	isoclinic class of #114713
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,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, 21917);