#183568

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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 ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

668 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:

(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);(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(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);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(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);(3,9)(6,12)(10,15)(13,16)(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,d^a=dh,e^b=eh,f^b=fh,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90764);