#482833

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
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×#127	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#559 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

774 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:

(2,7,9,13)(8,14,16,15)(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);(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,16)(12,14)(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);(1,3,6,11)(2,8,9,16)(4,12,5,10)(7,14,13,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,39)(26,40)(27,38)(29,43)(30,31)(34,45)(35,44)(37,46)(41,48)(42,47);(1,4,6,5)(2,7,9,13)(3,10,11,12)(8,15,16,14)(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=h,d^2=h,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,d^c=dhi,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, 390029);