#118444

order	512
nilpotency class	3
isoclinism	isoclinic class of #118439
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

541 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:

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

Polycyclic presentation:

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

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, 25640);