#506575

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
prime radical	#13341	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 652 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=eh,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, 413771);