#506498

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

812 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,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, 413694);