#479804

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

779 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=ef,d^2=i,g^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,d^b=def,e^b=ei,f^b=fi,d^c=de,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387000);