#243123

order	512
nilpotency class	3
isoclinism	isoclinic class of #243115
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)(3,13,16,27)(4,9)(5,10)(6,11)(7,12)(8,15,23,21)(14,29,17,28)(18,24)(19,25)(20,26)(22,30,32,31)(33,34,37,42,39,43,38,54)(35,44,47,56,46,60,48,40)(36,49,53,58,51,59,52,41)(45,55,50,64,62,63,61,57);(1,3)(2,8)(4,17)(5,15)(6,21)(7,16)(9,22)(10,13)(11,27)(12,23)(14,19)(18,31)(20,30)(24,29)(25,32)(26,28)(33,35)(34,40)(36,50)(37,48)(38,47)(39,46)(41,57)(42,60)(43,56)(44,54)(45,53)(49,63)(51,61)(52,62)(55,59)(58,64);(1,4,6,18,7,19,5,20)(2,9,11,24,12,25,10,26)(3,14,15,31,16,17,21,30)(8,22,13,28,23,32,27,29)(33,36,37,52,39,51,38,53)(34,41,42,59,43,58,54,49)(35,45,47,61,46,62,48,50)(40,55,44,57,56,63,60,64);(1,5,7,6)(2,10,12,11)(3,15,16,21)(4,18,19,20)(8,13,23,27)(9,24,25,26)(14,30,17,31)(22,29,32,28)(35,46)(36,51)(40,56)(41,58)(44,60)(47,48)(49,59)(52,53)

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=di,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, 150319);