#411821

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	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2102 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)(11,23)(12,18)(13,17)(14,21)(15,20)(16,22)(19,26)(24,33)(25,32)(27,31)(28,36)(29,35)(30,37)(34,39)(38,40);(1,2,3,4)(9,11,15,25,16,26,14,13)(10,17,21,19,22,32,20,23)(12,27,29,39,30,38,28,24)(18,33,36,40,37,34,35,31);(6,8)(9,12)(10,18)(11,24)(13,27)(14,29)(15,28)(16,30)(17,31)(19,34)(20,36)(21,35)(22,37)(23,33)(25,38)(26,39)(32,40);(9,13,14,26,16,25,15,11)(10,19,20,17,22,23,21,32)(12,27,29,39,30,38,28,24)(18,34,36,31,37,33,35,40)

Polycyclic presentation:

2^9|b^2=ei,d^2=h,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, 319017);