#104570

order	512
nilpotency class	3
isoclinism	isoclinic class of #104565
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

527 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgi,c^b=cf,d^b=dg,e^b=eghi,f^b=fh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11766);