#406101

order	512
nilpotency class	3
isoclinism	isoclinic class of #395586
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14924
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 313297);