#397217

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1513 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,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, 304413);