#474283

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

806 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efgh,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381479);