#419185

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1924 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)(9,10,13,20,15,17,14,19)(11,21,22,16,25,31,24,33)(12,26,28,35,30,18,29,36)(23,34,27,32,39,37,38,40);(5,6)(7,8)(9,11,13,22,15,25,14,24)(10,16,19,21,17,33,20,31)(12,27,28,39,30,38,29,23)(18,34,35,40,26,37,36,32);(2,4)(6,8)(10,17)(11,22)(13,14)(16,31)(18,26)(21,33)(23,38)(24,25)(27,39)(28,29)(32,34)(37,40);(9,12,15,30)(10,18,17,26)(11,23,25,39)(13,29,14,28)(16,32,33,40)(19,36,20,35)(21,37,31,34)(22,38,24,27)

Polycyclic presentation:

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

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, 326381);