#113828

order	512
nilpotency class	3
isoclinism	isoclinic class of #113815
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

732 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:

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

Polycyclic presentation:

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

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, 21024);