#417467

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35659	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2452 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)(9,10,14,18)(11,19,22,33)(12,16,23,31)(13,25,28,17)(15,26,30,21)(20,36,37,35)(24,34,27,32)(29,39,40,38);(1,2)(3,4)(5,6)(7,8)(12,23)(16,31)(20,37)(24,27)(29,40)(32,34)(35,36)(38,39);(6,8)(9,11)(10,15)(12,20)(13,26)(14,22)(16,29)(17,33)(18,30)(19,25)(21,28)(23,37)(24,39)(27,38)(31,40)(32,35)(34,36);(2,4)(9,12,14,23)(10,16,18,31)(11,20,22,37)(13,27,28,24)(15,29,30,40)(17,34,25,32)(19,35,33,36)(21,39,26,38)

Polycyclic presentation:

2^9|a^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=dei,f^c=fi,h^c=hi,f^d=fi

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, 324663);