#317183

order	512
nilpotency class	3
isoclinism	isoclinic class of #317171
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

628 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 64

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,d^c=di,f^c=fi,g^c=gh

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, 224379);