#16133

order	256
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁴
prime radical	C₂⁵	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=eg,b^2=eh,d^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

maximal subgroup of	C₂×#16133, #610231, #610274, …
maximal normal subgroup of	C₂×#16133, #610231, #610274, …
maximal quotient of	#154821, #154831, #154853, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 9121);