#477269

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

878 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)(3,7,8,6)(10,16)(13,14)(17,21)(18,31)(19,20)(22,37)(23,24)(26,36)(28,30)(32,40)(33,34)(35,38);(2,5)(6,7)(9,10,15,20)(11,21,25,33)(12,26,29,31)(13,19,14,16)(17,23,34,24)(18,30,36,28)(22,38,37,32)(27,35,39,40);(1,3,4,8)(2,6,5,7)(9,11)(10,17)(12,27)(13,24)(14,23)(15,25)(16,21)(18,35)(19,33)(20,34)(22,28)(26,32)(29,39)(30,37)(31,38)(36,40);(3,8)(6,7)(9,12)(10,18)(11,22)(13,30)(14,28)(15,29)(16,31)(17,32)(19,26)(20,36)(21,35)(23,39)(24,27)(25,37)(33,40)(34,38)

Polycyclic presentation:

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

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, 384465);