#136539

order	512
nilpotency class	4
isoclinism	isoclinic class of #12366
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1629
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8924
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dgi,e^c=eh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214124, #72214308
maximal normal subgroup of	#72214124, #72214308
maximal quotient of	#72216480, #72225257

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43735);