#148247

order	512
nilpotency class	4
isoclinism	isoclinic class of #148246
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11087
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,f^b=fh,h^b=hi,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55443);