#118641

order	512
nilpotency class	3
isoclinism	isoclinic class of #118637
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

542 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,f^b=fhi,d^c=di,e^c=ehi,f^c=fh,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, 25837);