#12915

order	256
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	Aut(G) = C₄×#7622880 (order 2048) · Inn(G) = #120 · Out(G) = C₂×C₄×D₄
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#120
commutator	C₂³	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1850
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#135946, #135947, #135956, …
maximal normal subgroup of	#135946, #135947, #135956, …
maximal quotient of	#135812, #135825, #135837, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5903);