#349977

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

828 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 164 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ei,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,c^b=ci,f^c=fh,g^c=gh,f^d=fh,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257173);