#482455

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
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×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

859 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,27)(20,32)(21,36,37,25)(22,26)(23,30,40,29)(24,35)(28,44)(31,39)(33,41,34,42)(38,47,48,45)(43,46);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,23,26,40)(20,28)(21,29)(22,31)(24,38,32,48)(25,39,36,27)(30,37)(33,45)(34,47)(35,46)(41,43,42,44);(2,7)(6,14)(8,15)(9,16)(17,20,22,35)(18,24,26,32)(19,28,31,46)(21,34,37,33)(23,38,40,48)(25,42,36,41)(27,43,39,44)(29,47,30,45);(17,21)(18,25)(19,29)(20,33)(22,37)(23,39)(24,41)(26,36)(27,40)(28,45)(30,31)(32,42)(34,35)(38,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=hi,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=di,f^d=fi,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, 389651);