#200168

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60953	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

983 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107364);