#465842

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,10,15,9,14,11,24)(7,16,20,17,19,12,25,28)(8,21,22,13,23,29,26,30)(18,32,33,27,35,36,34,31);(1,2,3,4)(5,7,11,25,9,19,10,20)(6,12,24,17,14,16,15,28)(8,18,26,34,23,35,22,33)(13,27,21,32,30,31,29,36);(2,4)(5,6)(7,17)(8,13)(9,14)(10,24)(11,15)(12,25)(16,20)(18,32)(19,28)(21,22)(23,30)(26,29)(27,34)(31,33)(35,36);(5,8,10,22,9,23,11,26)(6,13,15,29,14,30,24,21)(7,18,20,33,19,35,25,34)(12,27,28,36,16,31,17,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373038);