#506576

order	512
nilpotency class	4
isoclinism	isoclinic class of #501524
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 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 673 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:

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

Polycyclic presentation:

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

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, 413772);