#120943

order	512
nilpotency class	3
isoclinism	isoclinic class of #10496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

431 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28139);