#412532

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319728);