#475761

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1344 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 202 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)(7,13)(8,22)(9,28)(10,17)(11,16)(12,18)(14,25)(15,21)(19,31)(20,30)(23,32)(24,29)(26,33)(27,34)(35,36);(1,2,4,3)(5,7,11,9,12,21,10,20)(6,13,16,28,18,15,17,30)(8,23,26,35,27,24,25,19)(14,31,22,32,33,36,34,29);(2,3)(5,8)(6,14)(7,19)(9,24)(10,26)(11,25)(12,27)(13,29)(15,32)(16,34)(17,22)(18,33)(20,23)(21,35)(28,36)(30,31);(1,3,4,2)(5,9,11,21,12,20,10,7)(6,15,17,28,18,13,16,30)(8,24,25,35,27,23,26,19)(14,32,22,36,33,29,34,31)

Polycyclic presentation:

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

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, 382957);