#454374

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1867 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,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, 361570);