#411822

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×#3058	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2110 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)(11,19)(12,18)(13,31)(14,21)(15,20)(16,22)(17,24)(23,34)(25,33)(26,32)(27,39)(28,36)(29,35)(30,37)(38,40);(1,2,3,4)(9,11,14,24,16,25,15,13)(10,17,20,19,22,31,21,33)(12,26,28,23,30,27,29,38)(18,34,35,32,37,40,36,39);(6,8)(9,12)(10,18)(11,23)(13,27)(14,29)(15,28)(16,30)(17,32)(19,34)(20,36)(21,35)(22,37)(24,26)(25,38)(31,39)(33,40);(9,13,15,25,16,24,14,11)(10,19,21,17,22,33,20,31)(12,27,28,38,30,26,29,23)(18,34,35,32,37,40,36,39)

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,g^a=gi,h^a=hi,c^b=cg,e^c=ei,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, 319018);