#470742

order	512
nilpotency class	3
isoclinism	isoclinic class of #470742
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

788 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:

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

Polycyclic presentation:

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

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, 377938);