#404116

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=ei,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^c=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311312);