#435080

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,15,16,13)(18,23)(19,27)(20,21)(24,31)(25,26)(29,30);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,18,21,25,22,26,20,23)(19,28,27,31,30,32,29,24);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,19)(18,24)(20,27)(21,29)(22,30)(23,28)(25,32)(26,31);(3,8)(6,7)(11,16)(13,15)(18,25)(20,21)(23,26)(24,32)(27,29)(28,31)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342276);