#482900

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	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#554 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

776 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fghi,d^c=dh,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, 390096);