#118544

order	512
nilpotency class	3
isoclinism	isoclinic class of #118543
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

567 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)(3,9,11,15)(4,7)(5,8)(6,12,10,13)(14,16)(17,18)(19,29)(20,34,38,44)(21,27)(22,26)(23,28)(24,31)(25,37,42,36)(30,39,47,48)(32,40)(33,43)(35,45,46,41);(1,2)(3,10,11,6)(4,7)(5,8)(9,12,15,13)(14,16)(17,19,21,31,23,33,22,32)(18,24,26,29,28,40,27,43)(20,35,36,30,38,46,37,47)(25,41,34,39,42,45,44,48);(1,3)(2,6)(4,13)(5,11)(7,15)(8,10)(9,16)(12,14)(17,20,23,38)(18,25,28,42)(19,30,33,47)(21,36,22,37)(24,39,40,48)(26,34,27,44)(29,45,43,41)(31,46,32,35)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=ehi,f^c=fh,g^c=ghi

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, 25740);