#33458

order	256
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂³	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

323 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2,4,3)(5,6)(7,17)(8,13)(9,25)(10,30)(11,14)(12,19)(15,28)(16,27)(18,42)(20,49)(21,43)(22,37)(23,52)(24,55)(26,40)(29,41)(31,39)(32,45)(33,48)(34,57)(35,54)(36,51)(38,47)(44,64)(46,67)(50,63)(53,62)(56,60)(58,68)(59,65)(61,66);(5,7,11,21)(6,12,14,33)(8,18,22,44)(9,26,29,57)(10,31,27,20)(13,32,37,59)(15,39,38,49)(16,40,30,34)(17,41,43,25)(19,47,48,28)(23,53,50,68)(24,56,51,46)(35,60,61,67)(36,62,55,58)(42,63,64,52)(45,66,65,54);(1,3,4,2)(5,8,11,22)(6,13,14,37)(7,18,21,44)(9,23,29,50)(10,24,27,51)(12,32,33,59)(15,35,38,61)(16,36,30,55)(17,42,43,64)(19,45,48,65)(20,46,31,56)(25,52,41,63)(26,53,57,68)(28,54,47,66)(34,58,40,62)(39,60,49,67);(6,14)(8,22)(12,33)(15,38)(16,30)(17,43)(18,44)(23,50)(24,51)(25,41)(34,40)(39,49)(45,65)(46,56)(53,68)(54,66)

Polycyclic presentation:

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

maximal subgroup of	#150150, #152285, #152439, …
maximal normal subgroup of	#150150, #152285, #152439, …
maximal quotient of	#483506, #483795, #483830, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26446);