#254116

order	512
nilpotency class	3
isoclinism	isoclinic class of #254083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

780 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

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

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, 161312);