#457626

order	512
nilpotency class	3
isoclinism	isoclinic class of #454041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=f,d^2=i,f^2=i,g^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,g^b=gi,d^c=de,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, 364822);