#134567

order	512
nilpotency class	4
isoclinism	isoclinic class of #124708
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

302 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=i,d^2=i,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fghi,g^c=gi,f^e=fi

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, 41763);