#116786

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

573 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)(3,10)(4,14)(5,8)(6,9)(7,11)(12,15)(13,16)(17,18,22,26,23,27,21,37)(19,28,32,39,31,40,30,24)(20,25,35,42,36,44,34,43)(29,45,47,38,46,41,33,48);(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,19,21,30,23,31,22,32)(18,24,26,28,27,39,37,40)(20,33,34,46,36,47,35,29)(25,41,42,48,44,45,43,38);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,20)(18,25)(19,29)(21,34)(22,35)(23,36)(24,38)(26,42)(27,44)(28,41)(30,33)(31,46)(32,47)(37,43)(39,48)(40,45)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=i,f^2=hi,g^2=h,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=ei,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23982);