#121186

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	#1443
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

369 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,e^c=ehi,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, 28382);