#417653

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2420 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)(9,10,15,20)(11,21,25,34)(12,17,28,33)(13,19,14,18)(16,23,30,24)(22,35,38,40)(26,32,27,31)(29,36,39,37);(5,6)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,29)(18,34)(19,21)(20,30)(26,36)(27,37)(28,38)(31,40)(32,35)(33,39);(2,4)(6,8)(9,12,15,28)(10,17,20,33)(11,22,25,38)(13,26,14,27)(16,29,30,39)(18,31,19,32)(21,35,34,40)(23,36,24,37);(10,18)(11,23)(13,14)(16,30)(17,31)(19,20)(22,36)(24,25)(26,27)(29,39)(32,33)(37,38)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 324849);