#465854

order	512
nilpotency class	3
isoclinism	isoclinic class of #458129
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1040 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 373050);