#121500

order	512
nilpotency class	3
isoclinism	isoclinic class of #10038
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	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(6,13)(9,15)(17,18,22,26,23,27,21,38)(19,28,31,41,32,40,30,24)(20,33,37,42,36,46,35,25)(29,39,45,43,34,47,44,48);(1,2)(3,9,10,15)(4,8)(5,7)(6,11,13,16)(12,14)(17,19,21,30,23,32,22,31)(18,24,26,28,27,41,38,40)(20,34,35,45,36,29,37,44)(25,39,33,43,42,47,46,48);(1,3)(2,6,7,13)(4,11)(5,10)(8,9,14,15)(12,16)(17,20,21,35,23,36,22,37)(18,25,26,33,27,42,38,46)(19,29,31,45,32,34,30,44)(24,39,40,48,41,47,28,43)

Polycyclic presentation:

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

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, 28696);