#110500

order	512
nilpotency class	3
isoclinism	isoclinic class of #110476
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

608 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)(3,11,14,8)(4,6)(5,10)(7,9)(12,15,16,13)(17,18,21,28,23,26,22,27)(19,29,31,39,32,24,37,40)(20,33,35,25,34,41,36,42)(30,44,46,43,45,48,47,38);(1,3,7,13,5,14,6,15)(2,8,4,12,10,11,9,16)(17,19,22,31,23,32,21,37)(18,24,27,40,26,29,28,39)(20,30,36,46,34,45,35,47)(25,38,33,44,42,43,41,48);(1,4,7,10,5,9,6,2)(3,12,13,11,14,16,15,8)(17,20)(18,25,26,42)(19,30)(21,35)(22,36)(23,34)(24,38,29,43)(27,33,28,41)(31,46)(32,45)(37,47)(39,48,40,44)

Polycyclic presentation:

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

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, 17696);