#465032

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,5)(3,6,7,8)(9,10,14,19)(11,21,23,17)(12,15,20,16)(13,27,29,28)(18,31,34,30)(22,36,37,40)(24,26,25,33)(32,38,35,39);(2,5)(6,8)(9,11,15,24,14,23,16,25)(10,17,20,33,19,21,12,26)(13,22,30,38,29,37,31,39)(18,32,28,36,34,35,27,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,19,14,20,16,10)(11,21,24,33,23,17,25,26)(13,28,30,18,29,27,31,34)(22,32,38,40,37,35,39,36);(3,7)(6,8)(9,13)(10,18)(11,22)(12,27)(14,29)(15,30)(16,31)(17,32)(19,34)(20,28)(21,35)(23,37)(24,38)(25,39)(26,40)(33,36)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,g^c=gi,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, 372228);