#199523

order	512
nilpotency class	3
isoclinism	isoclinic class of #199522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

976 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)(4,18,21,26)(5,10)(6,11)(7,12)(8,16)(9,19,25,20)(14,28,30,22)(15,24)(17,23)(27,29,32,31)(33,34)(35,39,45,53)(36,46,48,40)(37,41)(38,42)(43,51)(44,52,49,55)(47,60,62,54)(50,56)(57,63)(58,59)(61,64);(1,3,5,16,7,17,6,15)(2,8,11,13,12,24,10,23)(4,14,19,31,21,30,20,29)(9,22,26,27,25,28,18,32)(33,35)(34,39,42,53)(36,43)(37,49)(38,45)(40,51,46,59)(41,55,56,52)(44,50)(47,61)(48,58)(54,64,60,63)(57,62);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,18)(11,26)(12,25)(13,27)(15,29)(16,31)(17,30)(23,32)(24,28)(33,36)(34,40)(35,43)(37,47)(38,48)(39,51)(41,54)(42,46)(44,57)(45,58)(49,61)(50,62)(52,63)(53,59)(55,64)(56,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,24,23)(9,18,25,26)(14,29,30,31)(22,27,28,32)(33,37,38,50)(34,41,42,56)(35,44,45,49)(36,47,48,62)(39,52,53,55)(40,54,46,60)(43,57,58,61)(51,63,59,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106719);