#238834

order	512
nilpotency class	3
isoclinism	isoclinic class of #238834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3245	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 971 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)(3,10)(4,6)(5,9)(7,11)(8,13)(12,15)(14,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,28)(27,32)(29,31);(2,6)(3,11)(4,13)(7,15)(8,9)(10,14)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(1,3,4,12,5,11,13,16)(2,7,8,15,9,10,6,14)(17,19)(18,23)(20,29)(21,28)(22,27)(24,32)(25,31)(26,30);(3,11)(7,10)(12,16)(14,15)(19,27)(23,31)(26,32)(28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146030);