#479810

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17260
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

624 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:

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

Polycyclic presentation:

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

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, 387006);