#416256

order	512
nilpotency class	3
isoclinism	isoclinic class of #396100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2414
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15220
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

883 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=h,e^2=i,g^2=i,h^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,g^c=gi

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, 323452);