#424416

order	512
nilpotency class	3
isoclinism	isoclinic class of #418492
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=ef,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,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, 331612);