#422941

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1920 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)(9,10,15,21)(11,22,26,33)(12,27,31,36)(13,20,14,19)(16,25,18,24)(17,29,37,30)(23,35,39,32)(28,34,38,40);(1,2,3,4)(9,11,13,24,15,26,14,25)(10,16,19,22,21,18,20,33)(12,28,29,39,31,38,30,23)(17,34,36,35,37,40,27,32);(5,6,7,8)(9,12,13,29,15,31,14,30)(10,17,19,36,21,37,20,27)(11,23,25,38,26,39,24,28)(16,32,33,40,18,35,22,34);(2,4)(9,11)(10,18)(12,23)(13,25)(14,24)(15,26)(16,21)(17,35)(19,22)(20,33)(27,40)(28,30)(29,38)(31,39)(32,37)(34,36)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fi,e^2=i,f^2=i,g^2=i,h^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,f^b=fi,g^b=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, 330137);