#403987

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2531 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)(8,13)(9,22)(10,25)(11,16)(12,19)(14,24)(15,26)(18,29)(20,31)(21,28)(23,30)(27,33)(32,34)(35,36);(5,7,11,8)(6,12,16,28)(9,18,24,20)(10,21,26,19)(13,25,17,15)(14,27,22,34)(23,33,36,32)(29,35,31,30);(2,4)(5,8,11,7)(6,13,16,17)(9,20,24,18)(10,19,26,21)(12,15,28,25)(14,29,22,31)(23,32,36,33)(27,30,34,35);(5,9)(6,14,16,22)(7,18)(8,20)(10,23)(11,24)(12,27,28,34)(13,29,17,31)(15,30,25,35)(19,32)(21,33)(26,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311183);