#501330

order	512
nilpotency class	4
isoclinism	isoclinic class of #501330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 694 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,e^2=i,f^2=hi,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, 408526);