#480707

order	512
nilpotency class	4
isoclinism	isoclinic class of #480690
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#537	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,e^b=ei,f^b=fgi,d^c=di,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387903);