#17301

order	256
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	Aut(G) order 327680
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

463 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^8|a^2=fg,b^2=eh,c^2=eg,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	#17301⋊C₅
Fitting subgroup of	#17301⋊C₅
maximal subgroup of	C₂×#17301, #845187, #845202, …
maximal normal subgroup of	C₂×#17301, #845187, #845202, …
maximal quotient of	#156565, #156606, #156634, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10289);