#468933

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

710 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:

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

Polycyclic presentation:

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

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, 376129);