#422917

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×#3154	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1940 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)(6,8)(9,10)(11,21)(12,26)(13,30)(14,20)(15,19)(16,24)(17,29)(18,28)(22,39)(23,34)(25,36)(27,33)(31,37)(32,35)(38,40);(1,2,3,4)(9,11,15,32,14,25,13,24)(10,16,30,36,20,35,19,21)(12,27,31,23,29,40,28,22)(17,34,37,33,26,39,18,38);(5,6,7,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,31)(16,33)(18,19)(20,26)(21,34)(23,25)(24,40)(27,32)(30,37)(35,38)(36,39);(2,4)(9,12)(10,18)(11,23)(13,31)(14,29)(15,28)(16,34)(17,19)(20,37)(21,33)(22,25)(24,40)(26,30)(27,32)(35,39)(36,38)

Polycyclic presentation:

2^9|b^2=ei,c^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,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, 330113);