#472871

order	512
nilpotency class	3
isoclinism	isoclinic class of #472870
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16839
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

825 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)(3,15)(4,17)(5,25)(6,13)(7,12)(8,14)(9,18)(10,23)(11,28)(16,27)(19,26)(20,32)(21,29)(22,31)(24,30);(1,3,7,18,8,20,6,19)(2,9,12,32,14,26,13,15)(4,21,22,11,24,25,23,16)(5,17,27,31,29,30,28,10);(1,4,6,22,8,24,7,23)(2,10,12,30,14,31,13,17)(3,16,18,25,20,11,19,21)(5,26,27,32,29,9,28,15);(1,5,6,28,8,29,7,27)(2,11,12,25,14,16,13,21)(3,17,19,10,20,30,18,31)(4,15,22,9,24,32,23,26)

Polycyclic presentation:

2^9|b^2=egh,c^2=e,d^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,h^c=hi,e^d=ei,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, 380067);