#512005

order	512
nilpotency class	4
isoclinism	isoclinic class of #511896
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

945 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 945 conjugacy classes of subgroups — too many to draw. Only its 180 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fhi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250058, #72250078, #72250101
maximal normal subgroup of	#72250058, #72250078, #72250101

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419201);