#200231

order	512
nilpotency class	3
isoclinism	isoclinic class of #200158
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60953	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fi,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, 107427);