#33855

order	256
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	Aut(G) = C₂×D₄×#442 (order 1024) · Inn(G) = D₄×D₈ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^8|a^2=h,b^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gh,d^b=de,e^b=eg,g^b=gh,f^c=fh,e^d=eg,g^d=gh

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#33855, #7623655, #7623663, …
maximal normal subgroup of	C₂×#33855, #7623655, #7623663, …
maximal quotient of	#491714, #491717, #491721, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26843);