#464242

order	512
nilpotency class	3
isoclinism	isoclinic class of #457090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1741 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)(7,12)(8,19)(9,11)(10,15)(13,22)(14,24)(16,28)(17,27)(18,31)(20,33)(21,30)(23,25)(26,35)(29,34)(32,36);(6,11,15,14)(7,16)(12,25,28,31)(13,26,19,30)(17,32)(18,23)(20,34)(27,29,36,33);(2,4)(5,7)(6,12)(8,20)(9,23)(10,16)(11,25)(13,29)(14,31)(15,28)(17,21)(18,24)(19,33)(22,34)(26,36)(27,30)(32,35);(5,8,9,21,10,22,24,35)(6,13,14,30,15,19,11,26)(7,17,18,20,16,32,23,34)(12,27,25,29,28,36,31,33)

Polycyclic presentation:

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

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, 371438);