#134629

order	512
nilpotency class	4
isoclinism	isoclinic class of #124718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

287 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 100 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)(5,6,10,14,9,13,23,30,11,16,26,34,24,15,35,33)(7,17,19,36,18,29,25,31,8,20,22,32,21,12,27,28);(1,2,3,4)(5,7,11,8)(6,12)(9,18,24,21)(10,22,26,19)(13,17)(14,31)(15,20)(16,29)(23,27,35,25)(28,34)(30,32)(33,36);(5,8,11,7)(6,12)(9,18,24,21)(10,25,26,27)(13,20)(14,32)(15,17)(16,29)(19,35,22,23)(28,33)(30,31)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=h,d^2=i,e^2=hi,f^2=ghi,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fgh,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, 41825);