#482523

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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

907 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,27)(20,24)(21,34,35,25)(22,26)(23,29,37,46)(28,41)(30,42)(31,39,32,38)(33,40)(36,44,48,43)(45,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,30)(18,23,26,37)(20,28,33,45)(21,29,35,46)(24,36,40,48)(25,27,34,42)(31,43,32,44)(38,47,39,41);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,33)(18,24,26,40)(19,28,30,45)(21,32,35,31)(23,36,37,48)(25,39,34,38)(27,41,42,47)(29,44,46,43);(17,21)(18,25)(19,29)(20,31)(22,35)(23,27)(24,38)(26,34)(28,43)(30,46)(32,33)(36,47)(37,42)(39,40)(41,48)(44,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,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, 389719);