#314602

order	512
nilpotency class	3
isoclinism	isoclinic class of #251791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2482
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

537 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 136 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)(9,10,14,19,15,20,13,18)(11,16,22,31,23,32,29,36)(12,24,28,17,26,35,27,34)(21,33,25,40,37,39,38,30);(9,11)(10,16)(12,25,26,38)(13,29)(14,22)(15,23)(17,33,34,39)(18,36)(19,31)(20,32)(21,28,37,27)(24,30,35,40);(1,2,4,6)(3,8,7,5)(9,12)(10,17)(11,21)(13,28)(14,27)(15,26)(16,30)(18,35)(19,24)(20,34)(22,38)(23,37)(25,29)(31,39)(32,40)(33,36);(9,11)(10,16)(12,21)(13,29)(14,22)(15,23)(17,30)(18,36)(19,31)(20,32)(24,39)(25,28)(26,37)(27,38)(33,35)(34,40)

Polycyclic presentation:

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

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, 221798);