#121507

order	512
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

537 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=ehi,f^b=fh,d^c=di

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, 28703);