#120443

order	512
nilpotency class	3
isoclinism	isoclinic class of #10508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1441
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

415 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:

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

Polycyclic presentation:

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

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, 27639);