#121820

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

390 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ei,f^b=fh,d^c=dh,e^c=eh,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, 29016);