#121554

order	512
nilpotency class	3
isoclinism	isoclinic class of #10212
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

420 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)(3,9)(4,13)(5,8)(6,12)(7,14)(10,16)(11,15)(17,18,22,26,23,27,21,37)(19,28,32,39,31,43,30,24)(20,25,36,40,35,41,34,47)(29,44,46,38,33,42,45,48);(1,2)(3,10,11,6)(4,7)(5,8)(9,12,15,16)(13,14)(17,19,21,30,23,31,22,32)(18,24,26,28,27,39,37,43)(20,33,34,46,35,29,36,45)(25,38,40,42,41,48,47,44);(1,3,4,12,5,11,14,16)(2,6,7,9,8,10,13,15)(17,20)(18,25)(19,29,31,33)(21,34)(22,36)(23,35)(24,38,39,48)(26,40)(27,41)(28,42,43,44)(30,45,32,46)(37,47)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28750);