#204022

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

521 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)(3,13,16,8,17,22,15,23)(4,9,18,24,20,25,19,26)(14,27,28,21,29,31,30,32)(33,34)(35,45)(36,50)(37,42)(38,43)(39,44)(40,48)(41,52)(46,60)(47,55)(49,58)(51,57)(53,56)(54,63)(59,62)(61,64);(1,3,6,16,7,17,5,15)(2,8,11,22,12,23,10,13)(4,14,18,28,20,29,19,30)(9,21,24,31,25,32,26,27)(33,35)(34,40)(36,46)(37,47)(38,48)(39,49)(41,54)(42,45)(43,58)(44,55)(50,59)(51,61)(52,62)(53,63)(56,64)(57,60);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,21)(10,26)(11,24)(12,25)(13,27)(15,30)(16,28)(17,29)(22,31)(23,32)(33,36,38,52,39,53,37,51)(34,41,42,50,44,57,43,56)(35,46,48,62,49,63,47,61)(40,54,45,59,55,60,58,64);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,18,20,19)(8,13,23,22)(9,24,25,26)(14,28,29,30)(21,31,32,27)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,53,52)(40,45,55,58)(41,50,57,56)(46,61,63,62)(54,59,60,64)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111218);