#418985

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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×#3010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2272 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:

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

Polycyclic presentation:

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

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, 326181);