#500301

order	512
nilpotency class	4
isoclinism	isoclinic class of #499946
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

778 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407497);