#501463

order	512
nilpotency class	4
isoclinism	isoclinic class of #501330
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×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 503 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) = 128

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

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

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,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, 408659);