#418968

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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×#3013	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2232 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326164);