#477834

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385030);