#254904

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
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

830 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 146 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)(9,10,14,18)(11,15,21,24)(12,16,13,17)(19,23,20,22)(25,26)(27,30)(28,31)(29,33)(32,37)(34,38)(35,40)(36,39);(2,5)(6,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,22,17,23)(25,27,29,35)(26,30,33,40)(28,36,37,34)(31,39,32,38);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(12,13)(17,18)(20,21)(22,23)(26,31)(28,37)(30,38)(32,33)(34,36)(39,40);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,19)(15,22,24,23)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,f^c=fh,g^c=ghi

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, 162100);