#355822

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

1151 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eg,e^2=i,f^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, 263018);