#411530

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

874 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=ef,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=ei,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, 318726);