#501465

order	512
nilpotency class	4
isoclinism	isoclinic class of #501332
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

513 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,28,27,14,19,30,10,16,20,12,9,17,7,15)(8,21,22,13,23,36,29,35,25,31,18,33,26,34,24,32);(1,2)(3,4)(5,7,9,20,10,19,27,11)(6,12,14,15,16,28,17,30)(8,22,23,29,25,18,26,24)(13,31,32,36,33,21,35,34);(2,4)(5,8,9,23,10,25,27,26)(6,13,14,32,16,33,17,35)(7,18,20,24,19,22,11,29)(12,21,15,34,28,31,30,36);(7,19)(11,20)(12,28)(15,30)(18,22)(21,31)(24,29)(34,36)

Polycyclic presentation:

2^9|a^2=fhi,b^2=e,c^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408661);