#243113

order	512
nilpotency class	3
isoclinism	isoclinic class of #243106
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	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

531 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)(35,40,47,56,48,57,46,55)(36,49,52,60,53,59,51,41)(45,58,50,63,61,64,62,54);(1,3,6,14)(2,7,10,21)(4,15,17,12)(5,18,19,13)(8,20,23,31)(9,24,25,11)(16,28,30,29)(22,27,32,26)(33,35)(34,40)(36,50)(37,47)(38,46)(39,48)(41,58)(42,56)(43,55)(44,57)(45,51)(49,63)(52,61)(53,62)(54,59)(60,64);(1,4,5,16,6,17,19,30)(2,8,9,22,10,23,25,32)(3,12,13,28,14,15,18,29)(7,20,11,26,21,31,24,27)(33,36)(34,41)(35,45)(37,51)(38,52)(39,53)(40,54)(42,59)(43,49)(44,60)(46,50)(47,62)(48,61)(55,58)(56,64)(57,63);(33,37,39,38)(34,42,44,43)(35,46,48,47)(36,51,53,52)(40,55,57,56)(41,59,60,49)(45,50,61,62)(54,58,63,64)

Polycyclic presentation:

2^9|a^2=f,b^2=h,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,d^b=di,e^b=eh,f^b=fh,g^b=gi,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150309);