#442151

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×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1009 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)(5,6,9,14,10,12,22,26)(7,11,17,23,18,25,15,29)(8,19,20,34,21,13,27,28)(16,31,32,36,33,24,30,35);(1,2)(3,4)(5,7,9,17,10,18,22,15)(6,11,14,23,12,25,26,29)(8,16,20,32,21,33,27,30)(13,24,28,35,19,31,34,36);(6,12)(7,15)(9,22)(11,23)(13,19)(16,30)(17,18)(20,27)(24,35)(25,29)(31,36)(32,33);(2,4)(5,8)(6,13,12,19)(7,16)(9,20)(10,21)(11,24,25,31)(14,28,26,34)(15,30)(17,32)(18,33)(22,27)(23,35,29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 349347);