#3484

order	128
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

950 conjugacy classes of subgroups; only the 432 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 432 normal subgroups are shown.

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

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 5 generators on 12 points:

(2,4)(5,6)(7,11)(8,9)(10,12);(6,9)(10,11);(1,2)(3,4);(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,12)(9,11)

Polycyclic presentation:

2^7|b^a=bf,c^a=cg,e^a=ef,d^b=df

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2216);