#482027

order	512
nilpotency class	4
isoclinism	isoclinic class of #481837
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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

873 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18,21,26,22,25,34,41)(19,27,29,42,30,23,35,37)(20,24,33,40,32,39,45,48)(28,38,43,47,31,36,44,46);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,30)(18,23,25,27)(20,31,32,28)(21,29,34,35)(24,38,39,36)(26,42,41,37)(33,44,45,43)(40,46,48,47);(2,8)(7,15)(9,10)(11,16)(17,20,22,32)(18,24,25,39)(19,28,30,31)(21,33,34,45)(23,36,27,38)(26,40,41,48)(29,43,35,44)(37,46,42,47);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(20,32)(24,39)(28,31)(33,45)(36,38)(40,48)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,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, 389223);