#380746

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1218 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287942);