#11497

order	256
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₂²⋊C₈
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,24,27,37,26,23,29,36,45,21,38,20,35)(19,30,33,46,32,25,40,42,34,44,48,41,39,43,31,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,34)(18,25,29,43)(20,31,37,40)(21,39,24,32)(22,33,36,48)(26,41,35,46)(27,44,38,30)(28,42,45,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,36,23,37,24,22)(18,26,38,28,29,35,27,45)(19,31,39,48,34,40,32,33)(25,41,30,42,43,46,44,47)

Polycyclic presentation:

2^8|a^2=e,b^2=f,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,d^b=dh,e^b=efh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#128516, #129671, #129680, …
maximal normal subgroup of	#128516, #129671, #129680, …
maximal quotient of	#124208, #124227, #124415, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4485);