#465855

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,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, 373051);