#418979

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,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, 326175);