#113745

order	512
nilpotency class	3
isoclinism	isoclinic class of #113742
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

738 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)(3,11,13,7)(4,8)(5,18)(6,9)(10,21)(12,22,30,29)(14,23,19,27)(15,32,31,24)(16,28)(17,25)(20,26)(33,34)(35,45,48,59)(36,49,52,57)(37,42)(38,43)(39,44)(40,47,55,53)(41,51,58,50)(46,54)(56,62)(60,63)(61,64);(1,3,6,13)(2,7,9,11)(4,15,17,31)(5,19,21,14)(8,24,25,32)(10,27,18,23)(12,20,30,16)(22,28,29,26)(33,35)(34,40)(36,46)(37,53)(38,47)(39,48)(41,56)(42,59)(43,45)(44,55)(49,60)(50,63)(51,61)(52,62)(54,58)(57,64);(1,4,6,17)(2,8,9,25)(3,12,13,30)(5,20,21,16)(7,22,11,29)(10,28,18,26)(14,31,19,15)(23,32,27,24)(33,36)(34,41)(35,46)(37,51)(38,50)(39,52)(40,54)(42,49)(43,57)(44,58)(45,60)(47,63)(48,62)(53,61)(55,56)(59,64)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,e^c=ei,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, 20941);