#512373

order	512
nilpotency class	4
isoclinism	isoclinic class of #511887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419569);