#289323

order	512
nilpotency class	3
isoclinism	isoclinic class of #289291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=ghi,f^c=fi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196519);