#402020

order	512
nilpotency class	3
isoclinism	isoclinic class of #396106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2459
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14829
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

759 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:

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

Polycyclic presentation:

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

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, 309216);