#113876

order	512
nilpotency class	3
isoclinism	isoclinic class of #113863
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

728 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:

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

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=fhi,c^b=cf,d^b=di,e^b=eh,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, 21072);