#13322

order	256
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#447
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

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

normal subgroups only

This group has 260 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13322, #498830, #498832, …
maximal normal subgroup of	C₂×#13322, #498830, #498832, …
maximal quotient of	#142239, #142251, #142287, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6310);