#17281

order	256
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	Aut(G) order 393216
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

571 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 159 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=g,b^2=egh,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#17281⋊C₃, #72266640, C₂×#17281⋊C₃, …
Fitting subgroup of	#17281⋊C₃
maximal subgroup of	#125867, #126026, #126138, …
maximal normal subgroup of	#125867, #126026, #126138, …
maximal quotient of	#156506, #156607, #156671, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10269);