#412437

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1394 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:

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

Polycyclic presentation:

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

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, 319633);