#411843

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3061	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2040 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)(11,18)(12,26)(13,28)(14,22)(15,21)(16,20)(17,24)(19,31)(23,32)(25,33)(27,40)(29,36)(30,35)(34,38)(37,39);(1,2,3,4)(9,11,14,24,15,12,16,25)(10,17,20,18,21,33,22,26)(13,23,29,37,30,27,31,38)(19,32,35,34,36,40,28,39);(6,8)(9,12)(10,18)(11,15)(13,27)(14,25)(16,24)(17,22)(19,34)(20,33)(21,26)(23,30)(28,32)(29,38)(31,37)(35,40)(36,39);(9,13,14,29,15,30,16,31)(10,19,20,35,21,36,22,28)(11,23,24,37,12,27,25,38)(17,32,18,34,33,40,26,39)

Polycyclic presentation:

2^9|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, 319039);