#62871

order	256
nilpotency class	2
isoclinism	isoclinic class of #62862
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

3465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^8|c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#62871⋊C₃, #72227545, #72266647, …
Fitting subgroup of	#62871⋊C₃
maximal subgroup of	#346483, #346493, #346503, …
maximal normal subgroup of	#346483, #346493, #346503, …
maximal quotient of	#7626361, #7626398, #7626399, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55859);