#134570

order	512
nilpotency class	4
isoclinism	isoclinic class of #124937
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1533
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 ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 309 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)(5,6,8,14,9,15,22,33,10,11,20,26,24,13,30,31)(7,16,17,34,18,12,27,28,19,29,35,32,23,25,21,36);(1,2,3,4)(6,11)(8,20)(12,25)(13,15)(16,29)(17,35)(21,27)(22,30);(5,7,10,19)(6,12,11,25)(8,21,20,27)(9,23,24,18)(13,29,15,16)(14,32,26,34)(17,22,35,30)(28,31,36,33)

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^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,f^b=fi,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, 41766);