#415765

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=efi,d^2=gh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,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, 322961);