#488833

order	512
nilpotency class	4
isoclinism	isoclinic class of #488679
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#62359	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

810 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=ghi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396029);