#479784

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17246
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

522 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386980);