#462611

order	512
nilpotency class	3
isoclinism	isoclinic class of #458129
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

712 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=e,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 369807);