#17183

order	256
nilpotency class	2
isoclinism	isoclinic class of #17082
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

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^8|a^2=fg,b^2=f,c^2=f,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

maximal subgroup of	C₂×#17183, #844967, #845004, …
maximal normal subgroup of	C₂×#17183, #844967, #845004, …
maximal quotient of	#156484, #156500, #156523, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10171);