#17307

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

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

507 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:

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

Polycyclic presentation:

2^8|a^2=ef,b^2=efgh,c^2=g,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	#17307⋊C₃, #72231842, #72231843, …
Fitting subgroup of	#17307⋊C₃
maximal subgroup of	#125834, #126187, C₂×#17307, …
maximal normal subgroup of	#125834, #126187, C₂×#17307, …
maximal quotient of	#156596, #156599, #156653, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10295);