#8782

order	256
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#117
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#382
prime radical	C₂³	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^8|a^2=f,b^2=e,c^2=d,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,f^b=fg,f^c=fg

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#146086, #146087, #146088, …
maximal normal subgroup of	#146086, #146087, #146088, …
maximal quotient of	#99283, #99284, #99285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1770);