#113945

order	512
nilpotency class	3
isoclinism	isoclinic class of #113932
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	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

755 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=di,e^c=ei,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21141);