#7514

order	256
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂×D₄×C₁₆ (order 256) · Inn(G) = D₄ · Out(G) = C₂×C₁₆
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₃₂	D₄
commutator	C₄	C₂×C₃₂
Frattini, cosocle	C₂×C₃₂	C₂²
socle	C₂	C₂²⋊C₃₂
prime radical	#41	C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 128 points:

(1,2,5,12,6,13,28,47,7,14,29,48,32,51,75,96,8,15,30,49,33,52,76,97,35,54,78,99,80,101,116,124,9,16,31,50,34,53,77,98,36,55,79,100,81,102,117,125,4,11,24,43,25,44,69,91,26,45,70,92,71,93,112,122)(3,17,19,56,20,57,62,103,21,58,63,104,64,105,109,126,18,41,59,46,60,87,106,94,61,88,107,95,108,120,127,123,22,42,27,72,65,89,73,113,66,90,74,114,110,121,115,128,23,10,37,38,67,39,82,84,68,40,83,85,111,86,118,119);(1,3)(2,10)(4,23)(5,27)(6,20)(7,21)(8,18)(9,22)(11,42)(12,46)(13,39)(14,40)(15,17)(16,41)(19,31)(24,59)(25,67)(26,68)(28,73)(29,74)(30,37)(32,64)(33,60)(34,65)(35,61)(36,66)(38,50)(43,56)(44,89)(45,90)(47,94)(48,95)(49,72)(51,86)(52,57)(53,87)(54,58)(55,88)(62,77)(63,79)(69,106)(70,107)(71,111)(75,115)(76,82)(78,83)(80,108)(81,110)(84,98)(85,100)(91,103)(92,104)(93,121)(96,123)(97,113)(99,114)(101,105)(102,120)(109,117)(112,127)(116,118)(119,125)(122,126)(124,128)

Polycyclic presentation:

2^8|a^2=d,c^2=h,d^2=e,e^2=f,f^2=g,g^2=h,b^a=bc,c^b=ch,d^b=dh

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#7514, #153432, #153586, …
maximal normal subgroup of	C₂×#7514, #153432, #153586, …
maximal quotient of	#94511, #94576, D₄⋊C₆₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 502);