#499694

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406890);