#183339

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₄⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,g^2=i,b^a=be,c^a=ci,d^a=dh,e^b=eh,f^b=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90535);