#507763

order	512
nilpotency class	4
isoclinism	isoclinic class of #499946
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

437 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

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

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, 414959);