#110969

order	512
nilpotency class	3
isoclinism	isoclinic class of #110956
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

577 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)(4,6)(5,9)(7,13)(8,12)(11,14)(15,16)(17,18)(19,23,28,36)(20,24)(21,33)(22,25)(26,34)(27,38,30,35)(29,44,43,37)(31,40)(32,46)(39,41)(42,47,45,48);(2,6,9,8)(3,11)(7,10,15,14)(13,16)(17,19,21,29,22,28,34,43)(18,23,26,37,25,36,33,44)(20,30,32,45,31,27,41,42)(24,38,39,48,40,35,46,47);(1,3,12,16,5,11,4,13)(2,7,6,14,9,15,8,10)(17,20,22,31)(18,24,25,40)(19,27,28,30)(21,32,34,41)(23,35,36,38)(26,39,33,46)(29,42,43,45)(37,47,44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

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, 18165);