#238940

order	512
nilpotency class	3
isoclinism	isoclinic class of #238935
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

966 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 128

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

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

Polycyclic presentation:

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

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, 146136);