#15467

order	256
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	Aut(G) order 393216
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁴
prime radical	C₂⁶	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1441 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1441 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^8|a^2=eg,b^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#15467⋊C₃, #72227883, #72260945, …
Fitting subgroup of	#15467⋊C₃
maximal subgroup of	#142568, C₂×#15467, #609053, …
maximal normal subgroup of	#142568, C₂×#15467, #609053, …
maximal quotient of	#154677, #154964, #155124, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 8455);