#118457

order	512
nilpotency class	3
isoclinism	isoclinic class of #118454
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25653);