#404257

order	512
nilpotency class	3
isoclinism	isoclinic class of #393921
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62845	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2377 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 2377 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=efi,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=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, 311453);