#10336

order	256
nilpotency class	3
isoclinism	isoclinic class of #10336
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#392
commutator, Frattini, cosocle	C₂⁵	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

783 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

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

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16);(5,8)(9,11)(10,13)(12,15)(14,16);(1,3)(2,5)(4,7)(6,8)(13,16)

Polycyclic presentation:

2^8|b^a=bd,c^a=ce,f^a=fgh,c^b=cf,e^b=eh,d^c=dg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3324);