#476708

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1037 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ei,e^2=i,f^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^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=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, 383904);