#495343

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1985
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

761 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,c^b=ch,e^b=ei,f^b=fi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402539);