#102889

order	512
nilpotency class	3
isoclinism	isoclinic class of #102889
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

728 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,5)(4,7)(6,8)(9,10)(11,13)(12,15)(14,16);(2,5)(3,8)(6,7)(17,18)(19,21)(20,23)(22,24);(10,13)(11,16)(14,15)(18,21)(19,24)(22,23)

Polycyclic presentation:

2^9|d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,f^b=fi,e^c=eh,f^c=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10085);