#411315

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1141 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318511);