#433464

order	512
nilpotency class	3
isoclinism	isoclinic class of #418863
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1494 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)(5,6,12,15,10,9,11,25)(7,13,18,29,19,17,26,28)(8,14,24,31,22,21,23,32)(16,33,20,30,34,27,35,36);(5,7)(6,13,9,17)(8,20)(10,19)(11,26)(12,18)(14,30,21,36)(15,29,25,28)(16,23)(22,35)(24,34)(27,32,33,31);(5,8,10,22)(6,14,9,21)(7,16,19,34)(11,24,12,23)(13,27,17,33)(15,32,25,31)(18,35,26,20)(28,36,29,30);(2,4)(5,9)(6,10)(7,17)(8,21)(11,25)(12,15)(13,19)(14,22)(16,33)(18,29)(20,36)(23,32)(24,31)(26,28)(27,34)(30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340660);