#442154

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

955 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 349350);