#13315

order	256
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

258 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,12,18)(7,19,24,50)(8,25,29,47)(9,30,34,45)(10,17,11,16)(13,36,40,21)(14,35,43,28)(15,32,46,33)(20,51,53,68)(22,44,23,49)(26,48,55,66)(27,58,61,42)(31,38,54,39)(37,63,64,57)(41,59,65,52)(56,62,67,60);(5,7,12,24)(6,13,18,40)(8,26,29,55)(9,31,34,54)(10,23,11,22)(14,41,43,65)(15,44,46,49)(16,39,17,38)(19,45,50,30)(20,35,53,28)(21,32,36,33)(25,37,47,64)(27,59,61,52)(42,66,58,48)(51,67,68,56)(57,62,63,60);(2,4)(5,8)(6,14)(7,20)(9,27)(10,35)(11,28)(12,29)(13,37)(15,42)(16,47)(17,25)(18,43)(19,48)(21,52)(22,55)(23,26)(24,53)(30,56)(31,57)(32,62)(33,60)(34,61)(36,59)(38,65)(39,41)(40,64)(44,51)(45,67)(46,58)(49,68)(50,66)(54,63)

Polycyclic presentation:

2^8|a^2=h,b^2=h,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#93773, #93775, #151179, …
maximal normal subgroup of	#93773, #93775, #151179, …
maximal quotient of	#142255, #142283, #142367, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6303);