#413814

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

1168 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,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, 321010);