#134566

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	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 330 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) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,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, 41762);