#241268

order	512
nilpotency class	3
isoclinism	isoclinic class of #233789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 218 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 148464);