#412435

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
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

1404 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)(11,17,25,31)(12,26,29,34)(13,20,14,19)(16,24,32,23)(18,27,35,28)(22,30,38,40)(33,36,39,37);(1,2,3,4)(9,11,13,23,15,25,14,24)(10,16,19,17,21,32,20,31)(12,22,27,36,29,38,28,37)(18,30,34,33,35,40,26,39);(6,8)(9,11)(10,17)(12,22)(13,23)(14,24)(15,25)(16,20)(18,33)(19,32)(21,31)(26,30)(27,36)(28,37)(29,38)(34,40)(35,39);(9,12,13,27,15,29,14,28)(10,18,19,34,21,35,20,26)(11,22,23,36,25,38,24,37)(16,30,17,33,32,40,31,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=hi,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,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319631);