#462317

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,g^c=gi,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, 369513);