#239024

order	512
nilpotency class	3
isoclinism	isoclinic class of #239017
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	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

845 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 845 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,f^c=fhi,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, 146220);