#121519

order	512
nilpotency class	3
isoclinism	isoclinic class of #9946
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28715);