#348500

order	512
nilpotency class	3
isoclinism	isoclinic class of #348478
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=di,f^c=fi,g^c=gh,f^d=fh

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, 255696);