#237896

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

868 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,6)(3,7,8,5)(11,16)(13,15)(17,18)(19,24)(20,32)(21,26)(22,27)(23,28)(25,34)(29,45)(30,39)(31,41)(33,38)(35,44)(36,43)(37,40)(42,46)(47,48);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,33)(21,30)(22,37)(23,31)(25,42)(26,39)(27,40)(28,41)(29,35)(32,38)(34,46)(36,47)(43,48)(44,45);(10,13)(11,16)(14,15)(17,20,23,36)(18,25,28,44)(19,29,31,46)(21,35,22,34)(24,38,41,48)(26,32,27,43)(30,47,37,33)(39,45,40,42);(17,21,23,22)(18,26,28,27)(19,30,31,37)(20,34,36,35)(24,39,41,40)(25,43,44,32)(29,33,46,47)(38,42,48,45)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,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, 145092);