#430403

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1254 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,d^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=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, 337599);