#243103

order	512
nilpotency class	3
isoclinism	isoclinic class of #243097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

530 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)(3,11,14,24)(4,8)(5,9)(6,10)(7,13,21,18)(12,27,15,26)(16,22)(17,23)(19,25)(20,29,31,28)(30,32)(33,34,37,42,39,44,38,43)(35,40,46,55,47,56,53,60)(36,48,51,59,52,58,50,41)(45,57,49,63,62,64,61,54);(1,3)(2,7)(4,15)(5,18)(6,14)(8,20)(9,24)(10,21)(11,25)(12,16)(13,19)(17,28)(22,31)(23,27)(26,32)(29,30)(33,35)(34,40)(36,49)(37,46)(38,53)(39,47)(41,57)(42,55)(43,60)(44,56)(45,50)(48,63)(51,62)(52,61)(54,58)(59,64);(1,4,5,17,6,16,19,30)(2,8,9,23,10,22,25,32)(3,12,13,28,14,15,18,29)(7,20,11,26,21,31,24,27)(33,36)(34,41)(35,45)(37,50)(38,51)(39,52)(40,54)(42,58)(43,48)(44,59)(46,61)(47,62)(49,53)(55,64)(56,63)(57,60);(4,16)(8,22)(12,15)(17,30)(20,31)(23,32)(26,27)(28,29)(33,37,39,38)(34,42,44,43)(35,46,47,53)(36,50,52,51)(40,55,56,60)(41,58,59,48)(45,61,62,49)(54,64,63,57)

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi

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, 150299);