#481869

order	512
nilpotency class	4
isoclinism	isoclinic class of #11799
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

877 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)(17,18,21,26,22,24,32,37)(19,23,29,34,27,35,41,39)(20,25,30,38,31,36,43,46)(28,33,42,44,40,45,48,47);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,23,24,35)(20,28)(21,29)(22,27)(25,33,36,45)(26,39,37,34)(30,42)(31,40)(32,41)(38,47,46,44)(43,48);(2,7)(6,14)(8,15)(9,16)(18,24)(19,27)(25,36)(26,37)(28,40)(29,41)(38,46)(42,48);(17,20,22,31)(18,25,24,36)(19,28,27,40)(21,30,32,43)(23,33,35,45)(26,38,37,46)(29,42,41,48)(34,44,39,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fghi,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, 389065);