#424422

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1071 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:

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

Polycyclic presentation:

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

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, 331618);