#33385

order	256
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	Aut(G) order 12288
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×D₈
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂	C₂²×#102
prime radical	#3584	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

557 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^8|a^2=f,e^2=gh,f^2=h,g^2=h,b^a=be,e^a=eg,g^a=gh,e^b=eg,g^b=gh,d^c=dh

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#33385⋊C₃
maximal subgroup of	#497406, #497407, #498172, …
maximal normal subgroup of	#497406, #497407, #498172, …
maximal quotient of	#481057, #481058, #481269, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26373);