#472872

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

813 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:

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

Polycyclic presentation:

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

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, 380068);