#148389

order	512
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#481
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,4)(5,6)(7,8)(10,16,18,33)(12,13)(14,30)(17,34,36,21)(19,38,28,20)(22,23)(24,32)(26,29)(27,31)(35,39,40,37);(1,2,3,4)(9,10,12,16,14,20,13,28,15,18,27,33,30,38,31,19)(11,21,22,17,24,35,23,37,25,34,29,36,32,40,26,39);(6,8)(9,11,15,25)(10,17,18,36)(12,26,27,23)(13,29,31,22)(14,32,30,24)(16,21,33,34)(19,35,28,40)(20,39,38,37)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=dhi,f^c=fh,h^c=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, 55585);