#355825

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13791
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263021);