#110153

order	512
nilpotency class	3
isoclinism	isoclinic class of #110146
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×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

571 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=i,f^2=i,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,e^c=ehi,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, 17349);