#147384

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6,10,16,11,17,32,46,12,18,33,47,9,15,30,44)(7,19,22,48,21,37,51,45,23,43,31,59,29,13,34,36)(8,14,26,40,28,42,58,67,27,41,57,66,25,39,55,64)(20,38,49,62,24,52,53,68,50,63,60,65,54,35,56,61);(5,7)(6,13)(8,24,27,54)(9,29)(10,31)(11,21)(12,23)(14,38,41,63)(15,43)(16,45)(17,19)(18,37)(20,28,50,25)(22,33)(26,56,57,53)(30,51)(32,34)(35,42,52,39)(36,47)(40,65,66,62)(44,48)(46,59)(49,55,60,58)(61,64,68,67);(1,2,3,4)(5,8,10,26,11,28,32,58,12,27,33,57,9,25,30,55)(6,14,16,40,17,42,46,67,18,41,47,66,15,39,44,64)(7,20,22,49,21,24,51,53,23,50,31,60,29,54,34,56)(13,35,36,61,19,38,48,62,37,52,45,68,43,63,59,65)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=di,f^b=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54580);