#355823

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	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1187 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,h^a=hi,c^b=ce,e^c=ei,f^c=fi,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, 263019);