#145119

order	512
nilpotency class	4
isoclinism	isoclinic class of #143196
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1619
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28247	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

361 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|c^2=di,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=cf,f^b=fh,h^b=hi,e^c=eg,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52315);