#414580

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

985 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321776);