#433973

order	512
nilpotency class	3
isoclinism	isoclinic class of #418818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15494
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

989 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:

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

Polycyclic presentation:

2^9|a^2=eg,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,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, 341169);