#118249

order	512
nilpotency class	3
isoclinism	isoclinic class of #118246
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

601 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:

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

Polycyclic presentation:

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

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, 25445);