#238161

order	512
nilpotency class	3
isoclinism	isoclinic class of #234028
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

916 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

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

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, 145357);