#349631

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=dh,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256827);