#120971

order	512
nilpotency class	3
isoclinism	isoclinic class of #10378
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
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

447 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)(11,20)(12,25)(13,29)(14,18)(15,19)(16,22)(17,27)(21,33)(23,32)(24,37)(26,36)(28,34)(30,35)(31,39)(38,40);(2,3,7,6)(9,11,13,23,15,22,14,24)(10,16,18,32,19,20,29,37)(12,26,27,38,28,39,30,21)(17,33,25,31,35,40,34,36);(1,3,4,2,5,6,8,7)(9,12)(10,17)(11,21,22,38)(13,30)(14,27)(15,28)(16,31,20,36)(18,34)(19,35)(23,39,24,26)(25,29)(32,33,37,40)

Polycyclic presentation:

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

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, 28167);